3C 351 is one of the most X-ray-quiet radio quasars (alpha(OX) approximately 1.6). We have observed 3C 351 with the ROSAT position sensitive proportional counter (PSPC) and find a complex X-ray spectrum which is not well reproduced by a power law plus low-energy cutoff model. Soft excess, partial covering, and ''warm absorber'' models can all produce acceptable fits, although only the warm absorber model gives typical values for the high-energy continuum slope. The alpha(OX) measured by using quasi-simultaneous ROSAT, MMT, and HST observations is in the range 1.5-1.6, significantly above the average of 1.37 (sigma approximately 0.15) for a complete sample of 33 3CR quasars. If the soft excess or partial covering models are correct, 3C 351 appears X-ray-quiet in the PSPC band because it has an extremely steep or flat intrinsic high-energy spectral slope. However, if the warm absorber model is correct, the quasar is intrinsically X-ray-quiet; the normalization of the intrinsic (unabsorbed) X-ray emission is unusually low relative to the optical luminosity. We investigate the properties of our warm absorber model in some detail. The apparently complicated behavior of the fit parameters may be understood by considering the effects of changing absorbing column and ionization parameter on intrinsic power-law spectra of different slopes.