The uniformly convergent cubic spline difference scheme for the problem $Ly=\varepsilon y''+q(x)y=f(x)$, $0<x<1$, $q(x)\geq q>0$, $y(0)=\alpha_0$, $y(1)=\alpha_1$ on a non-uniform mesh is derived. The scheme provides for the location of a larger number of points in the boundary layers, while the order of accuracy and the structure of the matrix remain the same as in the uniform grid.