A new ideal magnetohydrodynamic instability is presented. It is shown that linear incompressible Alfvénic disturbances can get over-reflected and exponentially amplified in compressible plasma flows. A simple and transparent stability criterion for a two-layer model is derived. The instability does not require a shear in the flow and may arise for rather moderate sub-Alfvénic flow speeds. It is therefore important to be aware of the Alfvén instability when dealing with laboratory or astrophysical plasma flows.