A technique for diagnosing the core temperature kT(core) and shell areal density rhoR(shell) of laser-imploded targets by measurement of the space-resolved, continuum x-ray emission is described in this work. An estimate of kT(core) is obtained from a fit to the space-resolved core x-ray spectrum. Observation of absorption by the cooler surrounding shell material is used to estimate rhoR(shell). For the x-ray emission from the core to be detected, its flux density must exceed that of the time-integrated emission from the ablation layer, making this technique particularly applicable to diagnosis of targets with low-Z ablators. In contrast to techniques that depend upon measurements of the reaction products (e.g., knock-ons, secondary-reaction products), which are limited to low shell-areal-density implosions (< 100 mg/cm2), space-resolved continuum-absorption spectroscopy should be applicable to the diagnosis of higher shell-areal-density implosions (> 100 mg/cm2). This technique has been successfully applied to the diagnosis of evacuated, polymer-coated, deuterated polystyrene (CD) shell implosions (surrogate-cryogenic implosions) performed with the University of Rochester's, 24-beam uv, OMEGA laser system. Space-resolved spectra of the target x-ray emission (approximately 1 to approximately 6 keV) were obtained with a grazing-incidence reflection microscope, dispersed by a transmission grating. Both a high-energy tail and absorption at low energies were observed in the x-ray spectra allowing estimates of kT(core) and rhoR(shell) to be obtained.