We search for pair-produced Dirac magnetic monopoles in 35.7
pb-1 of proton-antiproton collisions at s**(1/2) = 1.96 TeV with the Collider
Detector at Fermilab (CDF). We find no monopole candidates corresponding to
a 95% confidence-level cross-section limit sigma < 0.2 pb for a monopole
with mass between 200 and 700 GeV/c2. Assuming a Drell-Yan pair production
mechanism, we set a mass limit m > 360 GeV/c2.