We report on an observation of the youngest Crab-like supernova remnant, 3C 58, undertaken with the ROSAT high-resolution imager. The presence of a compact X-ray source at the center of the remnant, suggested by earlier observations, is confirmed, although aspect problems with the Observatory allow us to set only an upper limit on the size of the source of less than 10''. We describe an efficient new algorithm for conducting period searches on sparsely sampled data with a relatively small number of photons, and use it to set an upper limit on the pulsed fraction of this source of 50%. We show that the diffuse X-ray emission from the remnant is cospatial with the nonthermal radio emission and constrain the spectral parameters of the X-ray power-law fit. We go on to explore several models for the compact source, showing that Crab-like nonthermal pulses and thermal emission from the entire stellar surface are unlikely to explain the emission. Polar caps heated by magnetospheric discharges could be responsible for the observed X-rays. We conclude that the low X-ray pulsed fraction, inferred emitting area and temperature for the compact source, nebular luminosities in the X-ray and radio bands, and the known age can all be reconciled if P similar to 200 ms, P similar to 4000 x 10(-15) s s(-1), and B similar to 3 x 10(13) G, which would make this the highest magnetic field pulsar known.
