A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on $a$ $priori$ (sequentially) adapted meshes and study its convergence. The scheme on $a$ $priori$ adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which 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 of refinement iterations $K$ for improving the solution. To solve the discrete problems aimed at the improvement of 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, precisely, under the condition $N^{-1}=o\left(ε^{\nu}\right)$, where $N$ denotes the number of nodes in the spatial mesh, and the value $\nu=\nu(K)$ can be chosen arbitrarily small for suitable $K$.