In this fast growing technology world, where communications play a major rule for connecting people and machines together, the growth in wireless applications have caused an increasing demand for gaining access to the radio spectrum. However, the outdated spectrum utilization policies, imposed by the regulatory bodies in the past century, have caused the spectrum to look over-saturated. Recently, the concept of opportunistic spectrum access has been introduced as a tool to overcome the scarcity of the spectrum. The latter technology offers a tremendous potential to improve the utilization of the radio spectrum by implementing an efficient sharing of the licensed spectrum, whereby a secondary user may utilize the primary user's licensed band as long as its interference to the primary receiver remains below a tolerable level. In this paper, we investigate the capacity gains offered by this spectrum-sharing approach in Rayleigh fading environments. In particular, we derive the fading channel capacity of a secondary user subject to both average and peak received-power constraints at the primary's receiver. Considering both constraints, we derive the ergodic and outage capacities along with their optimum power allocation policies for Rayleigh flat-fading channel, and provide closed-form expressions for these capacity metrics. Furthermore, numerical simulations are conducted to corroborate our theoretical results.