We present and study a stabilized mixed finite element method for single-phase compressible flow through porous media. This method is based on a pressure projection stabilization method for multiple-dimensional incompressible flow problems by using the lowest equal-order pair for velocity and pressure (i.e., the ${\mathbf{P}}_1-P_1$ pair). An optimal error estimate in divergence norm for the velocity and suboptimal error estimates in the $L^2$-norm for both velocity and pressure are obtained. Numerical results are given in support of the developed theory.