Melt chocolate morsels with water in a double boiler. Add the sugar and mix well; cool slightly. Add the egg yolks and mix well. Cool slightly. Add the vanilla. Beat the egg whites until stiff and fold into the mixture.
Line the sides and bottom of a 9x5x2-¾-inch loaf pan with transparent wrap. Place a row of ladyfingers (unsplit), very close together, along the bottom of the pan. Sprinkle with ⅓ of the rum and pour on half of the chocolate mixture. Repeat. Finish with a third layer of ladyfingers sprinkled with rum. Chill for 24 hours.
To serve, turn upside down on a chilled platter; remove the plastic wrap carefully. Cover the sides and top with whipped cream. Spread the top with chocolate curls. Serve in slices.