This paper presents a new fully implicit procedure for the solution of the steady incompressible Navier-Stokes equations in primitive variables. The momentum equations are coupled with a Poisson-type equation for the pressure and solved using the Beam and Warming approximate factorization method. The present formulation does not require the iterative solution of the pressure equation at each time step. Thus, the major drawback of the pressure-Poisson approach, which made it prohibitively expensive for complex three-dimensional applications, is eliminated. Numerical solutions for the problem of the two-dimensional driven cavity are obtained using a non-staggered grid at Re = 100, 400, and 1000. All the computed results are obtained without any artificial dissipation. This feature of the present procedure demonstrates its excellent convergence and stability characteristics. Those characteristics result from the coupling of the pressure equation, which is elliptic in space, with the momentum equations.