The uniformly convergent quadratic spline difference scheme is derived for the problem: $Ly=-\varepsilon y''+q(x)y=f(x)$, $0<x<1$, $q(x)\geq0$, $y(0)=\alpha_0$, $y(1)=\alpha_1$. A non-uniform mesh which provides for the location of a larger number of points in the boundary layers Is used.