We present new limits on low mass accelerator-produced pointlike Dirac magnetic monopoles trapped and bound in matter surrounding the D0 collision region of the Tevatron at Fermilab (experiment E-882). In the context of a Drell-Yan mechanism, we obtain cross section limits for the production of monopoles with magnetic charge values of 1, 2, 3, and 6 times the minimum Dirac charge of the order of picobarns, some 100 times smaller than found in similar previous Fermilab searches. Mass limits inferred from these cross section limits are presented.