We address the interface between the liquid phase and the solid phase of a solidifying medium. We experimentally evidence a new instability of this solidification interface that occurs when the liquid phase involves flows. This hydrodynamic instability is oscillatory. It generates two kinds of traveling waves which both periodically modify the growth conditions of the solid phase. This results in repetitive band-like modulations of the structure of the solidified material. This instability is expected to widely occur in freezing media due to the usual occurrence of convective flows.