When a spacetime has boundaries, the entangling surface does not have to be necessarily compact and it may have boundaries as well. Then there appear a new, boundary, contribution to the entanglement entropy due to the intersection of the entangling surface with the boundary of the spacetime. We study the boundary contribution to the logarithmic term in the entanglement entropy in dimensions d=3 and d=4 when the entangling surface is orthogonal to the boundary. In particular, we compute a boundary term in the entropy of N=4 supergauge multiplet at weak coupling. For gauge fields we use a prescription which is consistent with the positive area law. The boundary term is compared with the holographic calculation of the entropy based on the Ryu-Takayanagi proposal adapted appropriately to the present situation. We find a complete agreement between these two calculations provided the boundary conditions imposed on the gauge multiplet preserve 1/2 of the supersymmetry and the extension of the boundary into the anti–de Sitter bulk is a minimal hypersurface.