This paper presents a new wall model to compute turbulent boundary layers using the RANS equations in high Reynolds number flows. The model solves a two-point boundary value problem for a coupled set of equations for the streamwise velocity and the turbulent viscosity. Since it includes both the pressure gradient and the momentum balance of the full Navier-Stokes system, the ODE is valid farther from the wall. We implement the model within a Cartesian cut cell method, and use one-dimensional linelets in each cut cell to avoid the excessive mesh refinement that would otherwise be needed. The linelets are coupled to the outer Cartesian grid in a fully conservative manner, with two-way interaction between the linelets and the background grid. Detailed comparisons of velocity and eddy viscosity with three well-studied examples from the Turbulence Modeling Resource website are presented to demonstrate the model’s performance in two space dimensions, including an example with smooth-body separation. The results show the new model gives excellent results even when the y+ value of the first point is in the wake layer.