The mixed Dirichlet–Neumann problem for the Laplace equation in a connected plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of a domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and the boundary integral–equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero.