We report precise Doppler measurements of GJ 317 (M3.5 V) that reveal the presence of a planet with a minimum mass M_P sin i = 1.2 M_Jup in an eccentric, 692.9 day orbit. GJ 317 is only the third M dwarf with a Doppler-detected Jovian planet. The residuals to a single-Keplerian fit show evidence of a possible second orbital companion. The inclusion of a second Jupiter-mass planet (P ≈ 2700 days, M_P sin i = 0.83 M_Jup) decreases from 2.02 to 1.23, and reduces the rms from 12.5 to 6.32 m s^-1. A false-alarm test yields a 1.1% probability that the curvature in the residuals of the single-planet fit is due to random fluctuations, lending additional credibility to the two-planet model. However, our data only marginally constrain a two-planet fit, and further monitoring is necessary to fully characterize the properties of the second companion. To study the effect of stellar mass on giant planet occurrence, we measure the fraction of stars with planets in three mass bins comprised of our samples of M Dwarfs, solar-mass stars, and intermediate-mass subgiants. We find a positive correlation between stellar mass and the occurrence rate of Jovian planets within 2.5 AU. Low-mass K and M stars have a 1.8% ± 1.0% planet occurrence rate compared to 4.2% ± 0.7% for solar-mass stars and 8.9% ± 2.9% for the higher mass subgiants. This result indicates that the former F- and A-type stars with M_* ≥ 1.3 M in our sample are nearly 5 times more likely than the M dwarfs to harbor a giant planet. Our analysis shows that the correlation between Jovian planet occurrence and stellar mass exists even after correcting for the effects of stellar metallicity.