The purpose of this note is to give a complete proof of a C^{0,α} regularity result for the pressure for weak solutions of the two-dimensional “incompressible Euler equations” when the fluid velocity enjoys the same type of regularity in a compact simply connected domain with C^2 boundary. To accomplish our result we realize that it is compulsory to introduce a new weak formulation for the boundary condition of the pressure which is consistent with, and equivalent to, that of classical solutions.