This paper deals with a free boundary porblem connected with the concept “quadrature surface”. Let Ω⊂$R$^{$n$}be a bounded domain with a$C$^{2}boundary and μ a measure compactly supported in Ω. Then we say ∂Ω is a quadrature surface with respect to μ if the following overdetermined Cauchy problem has a solution. $$\Delta u = - \mu in \Omega ,u = 0 and \frac{{\partial u}}{{\partial v}} = - 1 on \partial \Omega .$$