“Christopher Swann is a uniquely talented individual who uses his skills as a sailor to communicate his love of nature. His enthusiasm often appears to be boundless and it is certainly infectious. A period of time spent at sea with Swanny is not to be missed, and, for those who are willing to open their eyes to the world around them, it will be an experience that will be repeated again and again”.

Since dawn we have been motoring in an oily calm across the Bay of La Paz. To the west lies the coast of Baja California, its cactus-covered slopes glowing in the morning light.

There is no wind, and in the ensuing silence when the engines are switched off we can hear the breathing of the great whales all around us. First the hollow rush of expelled air from their great lungs, then the deep intake of air before the next dive.

When they blow, sending columns of moisture 30ft into the sky, the spray drifts over us, smelling faintly of cabbage, and when they surface to peer at us with curious, myopic eyes, it is impossible not to feel one's senses reaching out to them.

Here we are - two species from alien worlds - drawn together by mutual curiosity. And who, watching them come so trustingly close, can fail to be moved, knowing how we have persecuted these innocent beasts down the centuries?

Forget the great whales of Newfoundland, the right whales of Hermanus and Patagonia, the humpbacks of Alaska and the orcas of British Columbia.

For me the Sea of Cortez is the best place in the world to watch whales - not just for sheer numbers or variety of species but also for silky seas and a blissful desert climate. Yet hardly anyone seems to know about it. So quiet is the Baja, so empty and utterly unexploited, that you cannot believe Los Angeles and its roaring freeways are less than two hours away by air.

Cut off from the Pacific by a thousand miles of mountains, the Sea of Cortez resembles a giant marine oasis in the midst of the Mexican desert. Steinbeck wrote about it. Hemingway should have done. Otherwise it has managed to stay off the map.

The Baja itself is longer than Italy, a waterless peninsula hanging off the end of California with its tail in the Tropic of Cancer. Once you leave the main highway from Tijuana to Los Cabos you'll need a four-wheel-drive vehicle to get anywhere. Or a boat.

So, if you want to see whales galore, fly to LA and then down to La Paz, a laid-back ferry-port at the tip of the Baja. There, you sign on for a nine-day cruise with Christopher Swann, a latter-day Ahab who hunts whales not with a harpoon but with love in his heart and a pair of binoculars clapped to his eyes.

The Jacana, our live-aboard home for the next nine days, is a 45ft catamaran that Swann has chartered for the season.

She has four comfortable cabins and a spacious galley-cum-dining saloon in which Anne, Swann's sister, conjures up delicious feasts of char-grilled fish and enchiladas. Behind us we tow a panga, a fast fibreglass launch of the kind used by the local fishermen.

Within an hour of arriving at La Paz we are off into the blue, heading towards the island of Espiritu Santo, where we will anchor for the night. Above us, frigate birds patrol the skies and squadrons of pelicans lumber past.

The sun is shining, the sea is calm and Swann is in his element. "There she blows," he cries as a pillar of spray erupts ahead. It's a humpback, the first of 175 whales we shall see on this voyage.

Swanny, as he is known to his chums, is a footloose adventurer who should have been born a couple of centuries ago when there were still new trade routes to discover. Instead, he has become the Whale Whisperer of Cortez, seemingly able to summon up whales from the emptiest patch of water.

The last time we met was a decade ago, on his own boat, the Marguerite Explorer, a Danish trawler he had converted for wildlife cruises in the Hebrides.

Now, at the age of 47, he has swapped his Scottish oilskins for a salt-stained sarong, set up a tour company called Oceanus - based in Brighton of all places - and spends half the year chasing whales in the Baja sunshine.

Espiritu Santo is a typical Baja anchorage; an emerald cove rimmed with white shell sand, encircled by a Wild West landscape of red rim rocks and cactus forest.

Next morning, woken at dawn by the rattle of the anchor chain, I surface for a cup of tea to find we are already under way, with Swanny sitting barefoot at the wheel.

"Today," he announces, "we are looking for Moby." He means the sperm whale, the biggest of the toothed whales, which can grow up to 60ft long. "They're usually quite shy," he says, but finds one in half an hour.

It's a solitary bull, easily identified by its huge, blunt head and the way it lolls on the surface. When it dives, sliding back into its own weightless world of indigo currents and sonic visions, Swanny switches on the hydrophone and we listen to the barrage of clicks the whale emits as it hunts for squid.

We move on. Swanny is desperate to show us a blue whale, the biggest animal ever to grace our planet.

Just how big is biggest? Put it like this: a toddler could crawl through its aorta and you could drop a dog down its blowhole. Fully grown, a blue whale may weigh 190 tons and exceed 100ft in length; even at birth it is 25ft long. And it has the loudest voice in the animal kingdom; its low-frequency calls can be heard across oceans.

In the end, the blue whale finds us. We are floating far out in the bay, enjoying a picnic lunch on deck when we hear the familiar sound of a whale blowing and see the animal heading straight towards us. It is unbelievably huge - like a Polaris submarine.

Twice it circles us, no more than a boat's length away, and then it dives, passing directly beneath us. One flip of its tail could have sent us all to kingdom come; but there was no hint of menace in its unhurried movements.

During the 1960s, when whales were being slaughtered at the rate of 60,000 a year, it was feared that the giant blues were doomed. By 1970 no more than 8,000 were left. But by then the tide of international opinion was running in the whales' favour.

Even so, it took another 15 years before commercial whaling was banned. Today, whales are worth more alive than dead. Whale-watching holidays have become big business, worth in excess of $1bn (£588m) worldwide, and the blue whale has survived.

In the days that follow we see nine more blue whales, 79 fin whales, 66 humpbacks, 13 Bryde's whales, two sei whales and another couple of sperm whales.

Like Swanny, we have become adept at spotting the distant puffs of drifting spray, the sunlight gleaming on polished obsidian backs and the white shell-bursts of breaching humpbacks.

Yet even these are upstaged by our encounters with dolphins. Elsewhere I have seen maybe 30 at a time, but in the Sea of Cortez they turn up in their thousands. By the time we fly home we will have seen at least 23,000 common and bottlenose dolphins.

Joyfully they race towards us, determined to hitch a ride in our wake, and within minutes we are engulfed by a Mexican wave of leaping bodies.

They are chasing fish as they charge along, and they are not alone. All around us, boobies and pelicans are diving into the water. It's a feeding frenzy and we are at the heart of it, with thousands of dolphins strung out for a mile on either side of us.

Peering over the bows I can see them flying beneath us like torpedoes, their dorsal fins cleaving the surface a mere fingertip away. When one of them jumps ahead of us, soaring 10ft into the air and falling back with a crash that soaks us all, I suddenly become aware of this demented voice, hee-hawing like a cowboy, and realise to my horror that it is mine.

Then, as suddenly as they materialised out of the heat haze, they tire of their sport with us and peel away to resume their wild hunt, receding into the distance with a sound like breaking surf.

We, too, continue our voyage, heading north to Isla Coyote, where a score of fishermen, pangueros, live in bleached wooden shacks held fast like limpets beneath a simple white chapel.

We wade ashore past the horned skull of a steer with a gull perched on it and buy a yellow-fin tuna the men had caught that morning. I talk with one of them. His name is Ishmael and he wears a red T-shirt and a straw sombrero. The fishing is good, he tells me, but life is hard.

Our life, by comparison, was never sweeter. No longer pale-faced refugees from the British winter, we have kicked off our shoes and become ocean-going nomads. At anchor we snorkel among shoals of tropical fish or go beachcombing for cowries along sugar-white beaches.

At breakfast we gorge ourselves on luscious fresh pineapples; and in the evenings, as the islands turn red and gold in the dying light, we drink margaritas and watch the moon come up over the Sierra La Giganta.

No wonder Swanny calls it the Sea of Dreams. Who will ever believe I saw more than 20,000 dolphins? But it is true. The Sea of Cortez pulsates with life.

Sometimes, peering down into the Baja's sunlit depths, we spot squadrons of manta rays cruising past like Stealth bombers on a mission. And one memorable morning we go swimming with the friendly sea lions of Isla Partida, whose doe-eyed youngsters are so inquisitive that they peer into our face masks from just inches away.

Birds, too, throng these waters in numbers beyond counting. Flocks of black stormy petrels flutter among the wavelets like swallows hawking for insects over an English meadow; and every day we pass huge rafts of grebes and phalaropes, meet lipstick gulls with coral beaks and ospreys perched on cliff-top eyries.

Too soon the voyage is nearly over and I cannot bear the thought of leaving. But the highlight of the trip is still to come. It happens on our last night when we set out after supper to look for whales by moonlight. We jump into the panga and race out into the bay, leaving a fiery trail of bioluminescence in our wake.

Our luck is in. The moon is full and the sea is smooth as glass - what Swanny calls "the Silky". He kills the engine and in the silence of the moonglow we sit and listen to the deep sighs of spouting whales. There must be at least a score - fin whales, Bryde's and humpbacks - all leisurely circling us on the black-and-silver water.

Away to the south-west I can see the lights of La Paz, a reminder of that other universe from which we have come. But now and for a while yet, we remain in the company of the great leviathans, at peace with them and with ourselves in Mexico's magical Sea of Dreams.