We have studied flux-lattice melting in single crystals of YBa2Cu3O7 and Bi2.2Sr2Ca0.8Cu2O8 using a high-Q mechanical oscillator. In YBa2Cu3O7 with an applied magnetic field Hc^, flux-lattice melting occurs at Hc2 as in a conventional three-dimensional superconductor. However, for Hc^ flux-lattice melting occurs 3.2 K below Hc2. We believe that this is evidence for a transition into a vortex-liquid state similar to that seen in two-dimensional superconducting films. For Bi2.2Sr2Ca0.8Cu2O8 the effect is even more pronounced; with a bulk superconducting transition of 75 K, the flux-lattice melting in this material occurs in both orientations near 30 K.