Cooperative wireless multicast is investigated, in which a source sends multicast messages to a number of users. For a multicast message from the source, some users do not successfully receive the message (called unsuccessful users). For a successful user (who successfully receives the message from the source), we define its worst relaying channel gain as the smallest channel gain among its channel gains to all unsuccessful users. It is proposed that the successful user whose worst relaying channel gain is the highest among the worst relaying channel gains of all successful users is selected to serve as a relay. Considering that the channels in the system are independent but nonidentically distributed Rayleigh fading, we derive a closed-form outage probability expression for the proposed scheme. It is shown that the proposed scheme can achieve full diversity, and thus, having a larger number of users can improve the outage performance. Furthermore, we study the time allocation strategy that determines the durations used by the source and the selected relay for their transmissions, respectively. An approximate optimal time allocation is derived. In addition, we also investigate the case with relay selection error and the case with mixed Rayleigh/Rician fading. Simulation results verify the performance of the proposed cooperative multicast scheme and time allocation strategy.