The supernova remnant 3C 391 (G31.9 + 0.0) was observed using the ROSAT Position Sensitive Proportional Counter (PSPC). The PSPC image reveals centrally concentrated X-ray emission inside the radio shell, anticorrelated with the radio image. The northwestern half of the remnant has a strong radio shell and weak, centrally concentrated X-ray emission. In contrast, the southeastern half has strong central X-ray emission and weak radio emission. A spectral analysis combining PSPC and Einstein IPC data indicates that the X-ray emission arises from a thermal plasma that has not attained ionization equilibrium, though a nonthermal model can be rejected only with 90% confidence. There is evidence, al the 90% confidence level, of slightly enhanced abundances of the intermediate alpha-burning elements, Mg, Si, and S. The best-fit N-H is 2.4 x 10(22) cm(-2) and kT is 0.5 keV (T = 5.9 x 10(6) K). The centrally concenc trated morphology and the probable thermal origin of the X-ray emission along with the radio shell suggest that 3C 391 is similar to W44 and W28. There are spectral variations between the northwestern and southeastern parts of the remnant: either the northwest has higher N-H or the southeastern part has lower temperature, the former being more consistent with the X-ray morphology. The difference in N-H is large enough to account far the reduced X-ray surface brightness in the northwest, and the density in the northwest is higher than that in the southeastern shell. The X-ray data are consistent with the idea that the progenitor supernova has exploded just inside a molecular cloud, but close to the edge. The northwestern structure is created by the propagation of the supernova shock into the cloud while the enhanced central X-ray emission arises from evaporation of clumps from the edge of the cloud trapped within the remnant. The southeastern radio and X-ray emission represent a ''breakout'' of the shock into low-density, intercloud material.