Ethanol, the alcohol in cocktails, freezes at -114 C. High-quality freezers are typically set at -18 C, so putting a drink in the freezer or adding water-ice will not freeze the alcohol. Even dry ice, at -79 C will still leave the alcohol a liquid. So, if you really want to freeze the alcohol solid, liquid nitrogen is the best choice. Liquid nitrogen (LN2) boils at -196 C (77 K), and when thoroughly mixed into a drink, will freeze the drink, and boil away leaving nothing behind but frozen yumminess.
Geek note: Most drinks are not 200 proof - so the water-alcohol mixtures freeze
at a higher temperature than pure alcohol. Still, it's pretty darn cold.