Solidification controlled by vacancy diffusion is studied by Monte Carlo simulations of a two-dimensional Ising model defined by a Hamiltonian which models a thermally driven fluid-solid phase transition. The nonequilibrium morphology of the growing solid is studied as a function of time as the system relaxes into equilibrium described by a temperature. At low temperatures the model exhibits fractal growth at early times and crossover to compact solidification as equilibrium is approached.