A novel method of computing numerical solutions to the incompressible form of the Navier-Stokes equations is presented. It is based on the pressure correction approach, but employs a regular grid finite-volume variable arrangement instead of the usual staggered grid arrangement. The pressure equation is derived such that effects which promote the well-known checkerboard instability are not present. A relevant compatibility constraint on pressure is satisfied by Neumann boundary conditions obtained using a vector identity. Implemented in a second-order-accurate finite-volume code, the algorithm is used to compute several example problems for which good results are obtained.