Grease a cookie sheet or line it with parchment paper.
Place 5 ounces of the chocolate in a heatproof bowl set over a pan of simmering water. Stir until completely melted. Remove it from heat and stir in the remaining 5 ounces chocolate until thoroughly smooth. Spoon 1/2 teaspoon melted chocolate onto prepared pan to form a small disk. Dip the balloon into the disk of chocolate, coating about a third of the balloon. Place the dipped end of the balloon on the chocolate disk. Repeat with remaining chocolate and balloons. Place in the refrigerator to set, about 30 minutes.
In order to release the air from the balloon, pinch the balloon just under the knot and cut a small hole in the surface between fingers and knot. Very gradually release air; if air is released too quickly, the bowls may break. Carefully peel the deflated balloons from the chocolate bowls.