We analyze five years of PLANET photometry of microlensing events toward the Galactic bulge to search for the short-duration deviations from single lens light curves that are indicative of the presence of planetary companions to the primary microlenses. Using strict event selection criteria, we construct a well defined sample of 43 intensively monitored events. We search for planetary perturbations in these events over a densely sampled region of parameter space spanning two decades in mass ratio and projected separation, but find no viable planetary candidates. By combining the detection efficiencies of the events, we find that, at 95% confidence, less than 25% of our primary lenses have companions with mass ratio q=0.01 and separations in the lensing zone, 0.6-1.6 Einstein ring radii. Using a model of the mass, velocity and spatial distribution of bulge lenses, we infer that the majority of our lenses are likely M dwarfs in the Galactic bulge. We conclude that less than 33% of M-dwarfs in the Galactic bulge have Jupiter-mass companions between 1.5 and 4 AU, and less than 45% have 3 Jupiter-mass companions between 1 and 7 AU, the first significant limits on planetary companions to M-dwarfs. We consider the effects of the finite size of the source stars and changing our detection criterion, but find that these do not alter our conclusions substantially.
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