Carsharing has been considered as an effective means to increase mobility and reduce personal vehicle usage and related carbon emissions. In this paper, we consider problems of allocating a carshare fleet to service zones under uncertain one-way and round-trip rental demand. We employ a two-stage stochastic integer programming model, in the first stage of which we allocate shared vehicle fleet and purchase parking lots or permits in reservation-based or free-floating systems. In the second stage, we generate a finite set of samples to represent demand uncertainty and construct a spatialtemporal network for each sample to model vehicle movement and the corresponding rental revenue, operating cost, and penalties from unserved demand. We minimize the expected total costs minus profit and develop branch-and-cut algorithms with mixed-integer, rounding-enhanced Benders cuts, which can significantly improve computation efficiency when implemented in parallel computing. We apply our model to a data set of Zipcar in the BostonCambridge, Massachusetts, area to demonstrate the efficacy of our approaches and draw insights on carshare management. Our results show that exogenously given one-way demand can increase carshare profitability under given one-way and round-trip price differences and vehicle relocation cost whereas endogenously generated one-way demand as a result of pricing and strategic customer behavior may decrease carshare profitability. Our model can also be applied in a rolling-horizon framework to deliver optimized vehicle relocation decisions and achieve significant improvement over an intuitive fleet-rebalancing policy.