We investigate the convergence of difference schemes for the one-dimensional heat equation with the coefficient of the time derivative containing a Dirac delta distribution. An abstract operator method is applied for analyzing this equation. The convergence rate estimate of the order $\mathcal{O}$$(h)$ in a special discrete $\widetilde{W}^{2,1}_2$ Sobolev norm, compatible with the smoothness of the solution, is obtained.