Applications of two-dimensional (2D) perovskites have significantly outpaced the understanding of many fundamental aspects of their photophysics. The optical response of 2D lead halide perovskites is dominated by strongly bound excitonic states. However, a comprehensive experimental verification of the exciton fine structure splitting and associated transition symmetries remains elusive. Here we employ low temperature magneto-optical spectroscopy to reveal the exciton fine structure of (PEA)2PbI4 (here PEA is phenylethylammonium) single crystals. We observe two orthogonally polarized bright in-plane free exciton (FX) states, both accompanied by a manifold of phonon-dressed states that preserve the polarization of the corresponding FX state. Introducing a magnetic field perpendicular to the 2D plane, we resolve the lowest energy dark exciton state, which although theoretically predicted, has systematically escaped experimental observation (in Faraday configuration) until now. These results corroborate standard multiband, effective-mass theories for the exciton fine structure in 2D perovskites and provide valuable quantification of the fine structure splitting in (PEA)2PbI4.