Ion mobility and calorimetry measurements have been used to probe the nature of the phase transitions in gallium clusters with 29-55 atoms. While most clusters appear to undergo a first-order transition between solidlike and liquidlike phases, a few show the signature of melting without a significant latent heat. These transitions appear to be the finite size analogue of a second-order phase transition, and they presumably occur for some cluster sizes because their solidlike phase is amorphous.