Summary: Exact three-point difference scheme on an irregular grid for numerically solving boundary value problems for second order nonlinear ordinary differential equations with a singularity of the first kind is constructed. We have proved the existence and uniqueness of the solution of this scheme as well as have shown the convergence of the associated iteration method. In order to determine the coefficients and the right-hand side of the exact difference scheme at an arbitrary node of the grid, some auxiliary initial value problems on a small interval around this node must be solved. If these initial value problems are solved numerically by a one-step method of high-order accuracy, truncated three-point difference schemes of high-order accuracy results. The effectiveness of proposed difference schemes is illustrated by a numerical example.