In this paper, we build and analyze the stability and consistency of an explicit scheme for the compressible barotropic Euler equations. This scheme is based on a staggered space discretization, with an upwinding performed with respect to the material velocity only (so that, in particular, the pressure gradient term is centered). The velocity convection term is built in such a way that the solutions satisfy a discrete kinetic energy balance, with a remainder term at the left-hand side which is shown to be non-negative under a CFL condition. Then, in one space dimension, we prove that if the solutions to the scheme converge to some limit as the time and space step tend to zero, then this limit is an entropy weak solution of the continuous problem. Numerical tests confirm this theory, and show in addition (in 1D, and thus in absence of contact discontinuities) a first-order convergence rate.