In this paper, we analyze the performance of a broadcast packet in a VANET with the slotted ALOHA protocol where locations of vehicles are modeled by a one-dimensional Poisson point process. We consider the packet delivery probability under a broadcast delay constraint. Since the successful transmission of a broadcast packet is significantly affected by interferences at receivers which are spatially correlated, it is important to capture the spatial correlations properly in order to obtain an accurate expression of the packet delivery probability in a VANET. However, the exact analysis of the spatial correlations in interference is not mathematically tractable. In this paper we provide an accurate approximation of the spatial correlations in interference and derive the packet delivery probability with the help of the approximation. Numerical and simulation results are provided to validate our analysis and to investigate the performance of a VANET.