We present ROSAT soft X-ray spectra for the members of a complete sample of 13 core-dominated, flat radio spectrum sources. The sample comprises all radio sources from a flux-limited radio catalog (S5GHz > 1 Jy; Kuhr et al. 1981) which are north of delta = 70-degrees, at galactic latitudes b > 10-degrees, and have a flat radio spectrum between 1.4 and 5 GHz (alpha(r) < 0.5; f is similar to nu(-alpha)). The sources have already undergone much study at radio and optical wavelengths and are classified in broad terms as quasars (8 sources) and BL Lac objects (5 sources). We find mean X-ray power-law energy indices of alpha(x) = 0.59 +/- 0.19 for the quasars and 1.36 +/- 0.27 for the BL Lac objects (68 % confidence range for two parameters of interest as determined by a maximum likelihood method), supporting earlier Einstein Observatory results for heterogeneous samples of sources (Worrall & Wilkes 1990). A non-zero dispersion on alpha(x) is found for both the quasars and the BL Lac objects. The quasar X-ray spectra are harder than the interpolated spectral index between the optical and X-ray bands, alpha(ox), and they cluster tightly around < alpha(ox) - alpha(x) > congruent-to 0.6. In contrast, the BL Lac objects give < alpha(ox) - alpha(x) > congruent-to 0, but with a relatively large dispersion (sigma is similar to 0.5) which is similar to that on alpha(x). The BL Lac objects separate into a group of three sources with alpha(x) < 1.0 and two sources with alpha(x) > 1.7. When we incorporate published radio, mm, and optical measurements and compare the X-ray and broad-band spectral indices alpha(x), alpha(rx), alpha(mm,x), and alpha(ox), the most obvious difference between the quasar and BL Lac subsamples lies within the X-ray band. We have fitted the multi-wavelength data to inhomogeneous synchrotron-self-Compton models and find that, for the BL Lac objects with steep X-ray spectra, synchrotron emission can account for the radio to soft X-ray measurements, whereas the BL Lac objects with hard X-ray spectra and the quasars require significant Compton emission to model the spectral flattening indicated by alpha(x) < alpha(ox).