The subject of this paper is an error estimate of the order $h^{1/2}$ in the $L^2$-norm for an explicit, fully discrete numerical scheme that approximates smooth solutions of the barotropic compressible fluid flow equations in the multidimensional case. Assuming some a-priori estimates for the discrete solution we derive an error estimate using a technique based upon stability results due to Dafermos \cite{Da} and DiPerna \cite{Di}, which were originally formulated for systems of conservation laws.