It is well known that there is no analytical expression for the electrical capacitance of a cube, even though it has been claimed that one can compute this capacitance numerically to high precision. However, there have been some disparities between reported numerical results of the capacitance of the unit cube. In this article, the “walk on planes” (WOP) algorithm [M. L. Mansfield, J. F. Douglas, and E. J. Garboczi, Phys. Rev. E 64, 061401 (2001)] is used to compute the capacitance of the unit cube. With WOP, we remove the error from the ε-absorption layer commonly used in “walk on spheres” computations so that there is no inherent error introduced in these WOP computations except the intrinsic Monte Carlo sampling error of size O(N1/2). This WOP technique comes from the isomorphism, provided by probabilistic potential theory, between the electrostatic Dirichlet problem on a conducting surface, and the corresponding Brownian motion first-passage expectation. The numerical result we obtain with WOP, 0.660 678 2±1×10−7, supports the deterministic calculations by Read [F. H. Read, J. Comp. Phys. 133, 1 (1997)], the results by Given et al. [J. A. Given, J. B. Hubbard, and J. F. Douglas, J. Chem. Phys. 106, 3721 (1997)] and our previous results [C.-O. Hwang and M. Mascagni, J. Korean Phys. Soc. 42, L1 (2003); M. Mascagni and N. A. Simonov, J. Comp. Phys. (in press)]. Also, it is noted that the exact value conjectured by Hubbard and Douglas is inconsistent with our calculations, and that some other calculations fall outside our error bounds.