We extend the shell model Monte Carlo approach to heavy deformed nuclei using a new proton-neutron formalism. The low excitation energies of such nuclei necessitate low-temperature calculations, for which a stabilization method is implemented in the canonical ensemble. We apply the method to study a well-deformed rare-earth nucleus, 162Dy. The single-particle model space includes the 50-82 shell plus 1f_{7/2} orbital for protons and the 82-126 shell plus 0h_{11/2}, 1g_{9/2} orbitals for neutrons. We show that the spherical shell model reproduces well the rotational character of 162Dy within this model space. We also calculate the level density of 162Dy and find it to be in excellent agreement with the experimental level density, which we extract from several experiments.