Data compression algorithms remove redundant information from a file. The extent to which a file size is reduced is a measure of the entropy. Recently, it has been suggested to use this technique to find the entropy from a simulation of a physical system. Here, we apply this technique to estimate the entropy from Monte Carlo simulations of the hard sphere system. Numerical results compare well with the limited available entropy estimates from the laborious thermodynamic integration method, while this new algorithm is much faster. Our results show the phase transition by calculation of the entropy for a large number of densities. A common tangent method is used to find the coexistence densities for the fluid-solid phase transition. The upper density deviates from the established density from the literature, while the lower density compares very well.