A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is studied. The scheme is constructed using a majorant function for the singular component of the discrete solution that allows us to find a priori a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter ε, the step-size of a uniform mesh in x, and also by the required accuracy of the discrete solution and the prescribed number K of refinement iterations for the improvement of the solution. When we solve the discrete problems to improve the solution, we use uniform meshes on the subdomains. The error of the numerical solution depends weakly on the parameter ε. The scheme converges almost ε-uniformly. The advantage of this approach consists in the uniform meshes used.