This paper considers opportunistic gray-space primary-secondary spectrum sharing when the primary is a rotating radar. We assume that a secondary device is allowed to transmit when its interference does not exceed the radar’s tolerable level, probably because the radar’s directional antenna is pointing elsewhere, in contrast to current approaches that prohibit secondary transmissions if radar signals are detected at any time. The secondary system is a cellular system using the shared spectrum in some but not all of its cells; this may occur when the cellular system needs the shared spectrum to supplement its dedicated spectrum, or for a broadband hotspot service. It is shown that, even fairly close to the radar, extensive secondary transmissions are possible, although subject to interruptions as the radar rotates. For example, with 20% of the cells transmitting in the shared spectrum, on average the cellular system can achieve a data rate close to the one obtained in dedicated spectrum, even at less than 9% of the distance that secondary transmissions will not cause harmful interference in the radar’s main beam. By evaluating quality of service, it is shown that spectrum shared with radar is attractive for applications that generate much of the traffic on the Internet, including video streaming, peer-to-peer file sharing, downloads of large files, and web browsing, but not for an application sensitive to interruptions, like VoIP.