The first rigorous, variational results for the nonadiabatic (i.e., non-Born-Oppenheimer) ground states of the six isotopomers of the hydrogen molecule are reported. Ground-state energies in Hartrees are: H-2[-1.164 025 023 2] [this result was reported by us earlier in Phys. Rev. Lett. 83, 2541 (1999)], D-2[-1.167 168 78], T-2[-1.168 535 65], HD[-1.165 471 906], HT[-1.166 002 033], and DT[-1.167 819 642]. Expectation values for the kinetic and potential energies, the internuclear distance and the square of the internuclear distance, the virial coefficient, and the square of the energy gradient norm for the optimized wave functions are also reported. The calculations were performed with a direct nonadiabatic variational approach using a new diatomic correlated Gaussian basis set exponentially dependent on interparticle distances and including pre-exponential powers of the internuclear distance.