Audio authentication is a critical task in multimedia forensics demanding robust methods to detect and identify tampered audio recordings. In this paper, a new technique to detect adulterations in audio recordings is proposed by exploiting abnormal variations in the electrical network frequency (ENF) signal eventually embedded in a questioned audio recording. These abnormal variations are caused by abrupt phase discontinuities due to insertions and suppressions of audio snippets during the tampering task. First, we propose an ESPRIT-Hilbert ENF estimator in conjunction with an outlier detector based on the sample kurtosis of the estimated ENF. Next, we use the computed kurtosis as an input for a support vector machine classifier to indicate the presence of tampering. The proposed scheme, herein designated as SPHINS, significantly outperforms related previous tampering detection approaches in the conducted tests. We validate our results using the Carioca 1 corpus with 100 unedited authorized audio recordings of phone calls.