We present Advanced Satellite for Cosmology and Astrophysics (ASCA) observations of the γ-ray blazar PKS 0528+134, obtained at two separate epochs in 1994 August and 1995 March. These data represent the first measurement of the X-ray continuum emission of this source in the medium-hard X-ray band. Both ASCA spectra are consistent with a single power law with photon index Γ ~ 1.7-1.8 and column density NH ~ 5 × 1021 cm-2, higher than Galactic. The X-ray flux increased by a factor of 4 in ~7 months without appreciable change of the spectral shape. During the lower state of 1994 August, PKS 0528+134 was observed simultaneously in the optical, X-rays, and at γ-ray energies with EGRET. The γ-ray intensity is the faintest detected thus far in the source, with a steep spectrum (Γ ~ 2.7). The extrapolation of the X-ray continuum to the γ-ray range requires a sharp spectral break at ~1022 Hz.We discuss the radio through γ-ray spectral energy distribution of PKS 0528+134, comparing the low state of 1994 August with the flare state of 1993 March. We show that in PKS 0528+134, a nonnegligible contribution from the external radiation field is present and that, although synchrotron self-Compton scenarios cannot be ruled out, inverse Compton upscattering of thermal seed photons may be the dominant cooling process for the production of the high-energy continuum in this blazar.