A time dependent bidimensional Stefan problem with a nonlinear convection governed by a Navier-Stokes equation in the fluid phase is considered. The main result in the paper is the existence of a weak solution for this problem. To prove this, the author introduces an approximating problem with a penalty term acting on the fluid region. Some useful a priori estimates and the equicontinuity of the approximating solutions can also be established as in the case of the problem with a convection governed by a linear Stokes equation. A weak solution is thus obtained as a limit of the approximating solutions by various compactness arguments. The dimension \(N=2\) is essential in the proof of the equicontinuity.