This is a study of the security of the Coherent One-Way (COW) protocol for quantum cryptography, proposed recently as a simple and fast experimental scheme. In the zero-error regime, the eavesdropper Eve can only take advantage of the losses in the transmission. We consider new attacks, based on unambiguous state discrimination, which perform better than the basic beam-splitting attack, but which can be detected by a careful analysis of the detection statistics. These results stress the importance of testing several statistical parameters in order to achieve higher rates of secret bits.