We present spectroscopy for globular clusters (GCs) in the elliptical galaxy NGC 4365, obtained with the Low-Resolution Imaging Spectrograph on the Keck I telescope. Previous studies have shown that the optical color distribution of GCs in NGC 4365 lacks the bimodal structure that is common in globular cluster systems, showing only a single broad peak. Measurements of Balmer line indices (Hβ, Hγ, and Hδ) on the GC spectra support recent suggestions by Puzia et al. on the basis of optical and near-infrared photometry that some of the clusters in NGC 4365 are intermediate-age (2-5 Gyr) and metal-rich (-0.4<~[Z/H]<~0) rather than old (~10-15 Gyr) and metal-poor. We also find some genuinely metal-poor, old clusters, suggesting that the ages and metallicities of the two populations conspire to produce the single broad distribution observed in optical colors. Based on data obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration.