The isotropic Compton profilesJ(q) of H2+(2Σg+) and H2(1Σg+) are accurately evaluated in the impulse approximation by employing the exact Born-Oppenheimer wavefunction for H2+and both SCF and correlated (CI) wavefunctions expanded in elliptical coordinate basis functions for H2. For H2, the values ofJ(q), computed using the CI wavefunctions of Davidson and Jones and Hagstrom and Shull, agree well with those obtained from the LCAO-CI wavefunction of B. Liu. The discrepancy between theory and experimental data from X-ray scattering remains, but much better agreement is found with recent high-energy electron-scattering measurements. For H2+, the present results cast doubt on some results reported earlier for the momentum distribution. Experimental measurements are needed to enable comparison with the accurate values reported here for the Compton profile.