A boundary value problem is considered for a singularly perturbedparabolic convection-diffusion equation; we construct a finitedifference scheme on {\it a priori} (sequentially) {\it adapted}meshes and study its convergence. The scheme on {\it a priori}adapted meshes is constructed using a majorant function for thesingular component of the discrete solution, which allows us to find{\it a priori} a subdomain where the computed solution requires afurther improvement. This subdomain is defined by the perturbationparameter $\eps$, the step-size of a uniform mesh in $x$, and alsoby the required accuracy of the discrete solution and the prescribednumber of refinement iterations $K$ for improving the solution. Tosolve the discrete problems aimed at the improvement of thesolution, we use uniform meshes on the subdomains. The error of thenumerical solution depends weakly on the parameter $\eps$. Thescheme converges almost $\eps$-uniformly, precisely, under thecondition $N^{-1}=o\left(\eps^{\nu}\right)$, where $N$ denotes thenumber of nodes in the spatial mesh, and the value $\nu=\nu(K)$ canbe chosen arbitrarily small for suitable $K$.