Put the peach purée in the bottom of a mixing glass without ice. Slowly pour in the prosecco while gently pulling the purée up the side of the mixing glass to mix with the prosecco. Don't stir briskly or the prosecco will lose its eervescence. Strain into a champagne flute. Float the peach liqueur on top.
The Bellini was invented by Giuseppe Cipriani in 1948, at Harry's Bar in Venice. Originally the drink was made only for four months of the year, when the sweet white peaches used for the purée were in season. But when the Cipriani empire spread to New York City, Giuseppe's son, Arrigo, found a flash-frozen peach purée that he could use year-round. Unfortunately, the purées are not widely available to the public, so it's hard to make the original Bellini unless you have a food processor to prepare your own.