A 14-point difference operator is used to construct finite difference problems for the approxi- mation of the solution, and the first order derivatives of the Dirichlet problem for Laplace’s equations in a rectangular parallelepiped. The boundary functions ϕ j on the faces Γ j, j = 1, 2, ..., 6 of the parallelepiped are supposed to have pth order derivatives satisfying the Ho¨lder condition, i.e., ϕ j ∈ Cp,λ(Γ j), 0 < λ < 1, where p = {4, 5}. On the edges, the boundary functions as a whole are continuous, and their second and fourth order derivatives satisfy the compatibility conditions which result from the Laplace equation. For the error uh − u of the approximate solution uh at each grid point (x1, x2, x3), |uh − u| ≤ cρp−4(x1, x2, x3)h4 is obtained, where u is the exact solution, ρ = ρ(x1, x2, x3) is the distance from the current grid point to the boundary of the parallelepiped, h is the grid step, and c is a constant independent of ρ and h. It is proved that when ϕ j ∈ Cp,λ, 0 < λ < 1, the proposed difference scheme for the approximation of the first derivative converges uniformly with order O(hp−1), p ∈ {4, 5}.