The Zaremba boundary-value problem is a boundary value problem for Laplace-type partial differential operators acting on smooth sections of a vector bundle over a smooth compact Riemannian manifold with smooth boundary but with non-smooth (discontinuous) boundary conditions, which include Dirichlet conditions on one part of the boundary and Neumann ones on another part of the boundary. In [21] I studied the heat kernel asymptotics of Zaremba boundary value problem. The construction of the global parametrix of the heat equation is described in detail and the leading order heat kernel is computed explicitly. Some of the first non-trivial coefficients of the asymptotic expansion of the trace of the heat kernel are computed as well.