For a two-phase Stefan problem with flux boundary conditions considered by J. R. Gannon and M. Primicerio in [l], we prove existence using Schauder's fixed point theorem. The non- linear integral operator having the fixed point property is a monotone operator in the sense of L. Collatz [2]. Therefore, we can use Schauder's theorem to construct upper and lower bounds for the free boundary. Several numerical examples demonstrate that this procedure works.