Let x(w), w=u+iv ∈ B, be a minimal surface in ℝ3 which is bounded by a configuration 〈Γ, S〉 consisting of an arc Γ and of a surface S with boundary. Suppose also that x(w) is area minimizing with respect to 〈Γ, S〉. Under appropriate regularity assumptions on Γ and S, we can prove that the first derivatives of x(u, v) are Holder continuous with the exponent α=1/2 up to the “free part” of ∂B which is mapped by x(w) into S. An example shows that this regularity result is optimal.