It is proved that the exponentially fitted quadratic spline difference scheme derived in [5] and applied to the singularly perturbed two-point boundary value problem $$ \aligned &\varepsilon y''+p(x)y'=f(x),\quad 00. \endaligned $$ has the first order of uniform convergence on non-uniform mesh. In order to achieve the uniform first order accuracy the special "almost uniform mesh" which satisfies the condition $h_i=h_{i-1}+Mh_{i-1}\max(x_i,\varepsilon)$ was constructed. The results are illustrated by numerical experiments.