It is widely speculated that a Type Ib supernova is the explosion of a Wolf-Rayet star. We calculate the X-ray and gamma-ray signatures of models of that type, assuming all hard photons to have originated with Ni decay chains, in hopes of providing diagnostics of the exposed-core models of massive stars, which constitute one model of the Wolf-Rayet stars, calculated by Ensman and Woosley (1988). These provide the characteristic luminosity peak and light curve of Type Ib supernovae for helium-core masses between 4 and 6 M⊙. We compute gamma-ray line shapes and fluxes and the Comptonized X-ray continuum resulting from the decay of the radioactive 56Co and 57Co isotopes that are synthesized by the explosion of the presupernova star (the suggested Wolf-Rayet or post-Wolf-Rayet star) with a Monte Carlo transport code. The expansion velocity, the total mass of the ejecta, the radial mixing of radioactivity in that ejecta, and the 56Ni yield effect both the strength and the evolution of the hard radiation. With the anticipated launch of Gamma Ray Observatory, we can hope to detect Type Ib supernovae to distances of 3 Mpc and utilize the characteristics of the gamma lines and X-ray spectrum to distinguish between differing Type Ib supernova models and to address their suggested relationship to Wolf-Rayet stars.