The initial velocity dispersion of newborn stars is a major unconstrained aspect of star formation theory. Using near-infrared spectra obtained with the APOGEE spectrograph, we show that the velocity dispersion of young (1-2Myr) stars in NGC 1333 is 0.92+/-0.12km/s after correcting for measurement uncertainties and the effect of binaries. This velocity dispersion is consistent with the virial velocity of the region and the diffuse gas velocity dispersion, but significantly larger than the velocity dispersion of the dense, star-forming cores, which have a subvirial velocity dispersion of 0.5km/s. Since the NGC 1333 cluster is dynamically young and deeply embedded, this measurement provides a strong constraint on the initial velocity dispersion of newly formed stars. We propose that the difference in velocity dispersion between stars and dense cores may be due to the influence of a 70{mu}G magnetic field acting on the dense cores or be the signature of a cluster with initial substructure undergoing global collapse.