Let $\{X_n\}$ be a stationary sequence of random variables whose marginal distribution $F$ belongs to a stable domain of attraction with index $\alpha, 0 < \alpha < 2$. Under the mixing and dependence conditions commonly used in extreme value theory for stationary sequences, nonnormal stable limits are established for the normalized partial sums. The method of proof relies heavily on a recent paper by LePage, Woodroofe, and Zinn which makes the relationship between the asymptotic behavior of extreme values and partial sums exceedingly clear. Also, an example of a process which is an instantaneous function of a stationary Gaussian process with covariance function $r_n$ behaving like $r_n \log n \rightarrow 0$ as $n \rightarrow \infty$ is shown to satisfy these conditions.