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We present a fourth-order accurate projection method for solving the incompressible Navier-Stokes equations in two and three dimensions, on rectangular domains. For spatial discretization, finite volume stencils are derived for the operators of convection, diffusion and approximate projection, for both periodic and no-slip wall boundary conditions. Time integration is based on a six-stage, stiffly-accurate, L-stable, implicit-explicit, additive Runge-Kutta (ARK) method, which treats the non-stiff convection term explicitly and the stiff diffusion term implicitly. Velocity is projected onto the divergence-free constraint for each intermediate stage in the time integration scheme. The resulting Poisson- and Helmholtz-type linear systems are solved with an efficient multigrid algorithm. In the case of no-slip boundary conditions, higher-order projection methods typically have difficulties due to the Laplace-Leray commutator; in order to maintain stability obtain and fourth-order convergence in this case, it was found necessary to extract the pressure from velocity and rigorously enforce the divergence-free condition on the boundary in a multidimensional manner. Spatial and temporal accuracy are demonstrated with well-resolved benchmark test problems, showing that in $L_1$- and $L_2$ norms fourth-order convergence is achieved for velocity and third-order convergence for pressure. The methodology is enhanced by adaptive mesh refinement for additional power of resolving boundary layers.