Thanks to recent advances in deep learning, sophisticated generation tools exist, nowadays, that produce extremely realistic synthetic speech. However, malicious uses of such tools are possible and likely, posing a serious threat to our society. Hence, synthetic voice detection has become a pressing research topic, and a large variety of detection methods have been recently proposed. Unfortunately, they hardly generalize to synthetic audios generated by tools never seen in the training phase, which makes them unfit to face real-world scenarios. In this work we aim at overcoming this issue by proposing a new detection approach that leverages only the biometric characteristics of the speaker, with no reference to specific manipulations. Since the detector is trained only on real data, generalization is automatically ensured. The proposed approach can be implemented based on off-the-shelf speaker verification tools. We test several such solutions on three popular test sets, obtaining good performance, high generalization ability and high robustness to audio impairment.