An American in Paris is unlike any other musical on the London stage: sumptuously beautiful and heartfelt, it has a romantic pizzazz all of its own.

In adapting the 1951 musical for the stage, book writer Craig Lucas and director and choreographer Christopher Wheeldon have effectively created a new musical, which grounds the love of GI Jerry Mulligan for the Parisian waif Lise in the realities of post-war Paris. As their tender love grows and thrives, overcoming numerous obstacles, the show itself moves from gloom to glory. It is, as the Gershwin score assembled from his entire back catalogue might put it "S'wonderful."

That score is one of the show's chief assets, opening as it does with the "Concerto in F" and concluding with the famous "An American in Paris" ballet, swiftly followed by the heartfelt "They Can't Take That Away From Me". The songs provide an armature on which to hang the plot, which unfolds like a memory play through the narration of pessimistic Adam. It all just about hangs together.

But it is the design and the direction that give An American in Paris its unique texture and tone. Bob Crowley's designs, a fleet and gorgeous mixture of stage flats and projections (courtesy of 59 Productions Ltd) do more than set the scene, they create a mood. From the moment the show opens with a billowing French flag replacing the Nazi swastika, and the streets of Paris hove into view in black and white, scribbled drawings, that in turn build to a glorious new dawn, the design expresses meaning as well as place.

Within this frame, Wheeldon lets his action flow. The dialogue and spoken scenes have been much tightened since the show's premiere in Paris in 2014, but he still plays to his strengths. As a classical dance choreographer, he knows that dance is a language and in a succession of dazzling sequences, he lets it tell the story – whisking Lise and Jerry through their romance in danced duets rather than sung ones.

His steps and his conception have deep refinement, but allow the emotion to emerge naturally from the action, whether it is in a version of "I've Got Rhythm" that grows gradually, or in the final ballet which is full of the passion that Lise has begun to feel. For good measure, he throws in one big tap number, but it is the bold use of balletic idiom, mixed with a more casual musical style, that makes the choreography so striking.

His purpose and instinct are perfectly served by his stars. Although many of the parts have been recast for this British production – with Jane Asher making a striking appearance as a haughty Frenchwoman – and are all well-played, Leanne Cope and Robert Fairchild created the parts of Lise and Jerry on stage, and inhabit them fully. He is ballet royalty, a principal at New York City Ballet, yet with the sly, sexy instincts of a Broadway hoofer, soft-shoe shuffling with easy grace and burning up the stage when he jumps and turns. He sings well, too and brandishes a megawatt charm of which Gene Kelly would be proud.

Since Paris, Cope has grown into her role. She was always a graceful dancer, now she sings and acts with a quiet confidence, creating a fully-rounded portrait of a girl on the threshold of finding true love, trying to do the right thing by everyone around her, trying to forget her suffering. She gives a glorious show its gentle heart.

An American in Paris is currently booking at the Dominion Theatre until 30 September 2017.