In the upper half of the unit ball B⁺ = {|x| < 1, x₁ > 0}, let u and Ω (a domain in R₊ ⁿ = {x ∈ Rⁿ: x₁ > 0}) solve the following overdetermined problem: Δu = χΩ in B⁺, u = |∇u| = 0 in B⁺ / Ω, u = 0 on ∏ ∩ B, where B is the unit ball with center at the origin, χΩ denotes the characteristic function of Ω, ∏ = {X₁ = 0}, n ≥ 2, and the equation is satisfied in the sense of distributions. We show (among other things) that if the origin is a contact point of the free boundary, that is, if u(0) = |∇u(0)| = 0, then ∂Ω∩ B_r0 is the graph of a C¹-function over ∏ ∩ B_r0. The C¹-norm depends on the dimension and sup B⁺ |u|. The result is extended to general subdomains of the unit ball with C³-boundary.