We give new upper and lower bounds on the minimax sample complexity of differentially private mean estimation of distributions with bounded $k$-th moments. Roughly speaking, in the univariate case, we show that $n = Θ\left(\frac{1}{α^2} + \frac{1}{α^{\frac{k}{k-1}}\varepsilon}\right)$ samples are necessary and sufficient to estimate the mean to $α$-accuracy under $\varepsilon$-differential privacy, or any of its common relaxations. This result demonstrates a qualitatively different behavior compared to estimation absent privacy constraints, for which the sample complexity is identical for all $k \geq 2$. We also give algorithms for the multivariate setting whose sample complexity is a factor of $O(d)$ larger than the univariate case.