Poisson's equations in a cuboid are frequently solved in many scientific and engineering applications such as electric structure calculations, molecular dynamics simulations and computational astrophysics. In this paper, a fast and highly accurate algorithm is presented for the solution of the Poisson's equation in a cuboidal domain with boundary conditions of mixed type. This so-called harmonic surface mapping algorithm is a meshless algorithm which can achieve a desired order of accuracy by evaluating a body convolution of the source and the free-space Green's function within a sphere containing the cuboid, and another surface integration over the spherical surface. Numerical quadratures are introduced to approximate the integrals, resulting in the solution represented by a summation of point sources in free space, which can be accelerated by means of the fast multipole algorithm. The complexity of the algorithm is linear to the number of quadrature points, and the convergence rate can be arbitrarily high even when the source term is a piecewise continuous function.