In cognitive radio technology, secondary users may be granted access to the spectrum bands occupied by a primary user as long as the interference-power, inflicted on the primary receiver as an effect of the transmission of the secondary user, is deemed unharmful. In this paper, we assume that the successful operation of the primary user requires a minimum-rate to be guaranteed by its channel for a certain percentage of time, and obtain the interference-power constraint that is required to he fulfilled by the secondary user. We investigate the capacity gains offered by this spectrum-sharing approach when the input transmit power of the secondary user is limited. In particular, we assume that only partial channel information of the link between the secondary's transmitter and primary's receiver is available to the former, and derive lower bounds on the capacity of a Rayleigh flat-fading channel with different transmission techniques, namely, channel inversion and constant-power transmission. Closed-form expressions for these capacity metrics are provided, and numerical simulations are conducted to corroborate our theoretical results.