Containerless undercooling by electromagnetic levitation has been used to measure both growth velocity and grain size as a function of undercooling in Cu70Ni30and Cu69Ni30B1alloys. At a critical undercooling &Dgr;T* the temperature dependence of the growth velocity changes discontinuously. This is accompanied by a sudden drop of grain size in the solidified product by more than two orders of magnitude. The comparative investigations show that it is not a critical undercooling that initiates the grain refinement process, but rather a critical crystal growth velocity, which approximately equals the diffusive speed.