The partial regularity of the weak heat flow of harmonic maps from a Riemannian manifold M with boundary into general compact Riemannian manifold N without boundary is considered. It is shown that the singular set Sing(u) of the weak heat flow satisfies Hpn(Sing(u)) = 0,with n = dimensionM. Here Hpn is Hausdorff measure with respect to parabolic metricρ((x, t), (y, s)) = max{|x - y|, |t - s|}.