The applicability of process mining techniques hinges on the availability of event logs capturing the execution of a business process. In some use cases, particularly those involving customer-facing processes, these event logs may contain private information. Data protection regulations restrict the use of such event logs for analysis purposes. One way of circumventing these restrictions is to anonymize the event log to the extent that no individual can be singled out using the anonymized log. This article addresses the problem of anonymizing an event log in order to guarantee that, upon release of the anonymized log, the probability that an attacker may single out any individual represented in the original log does not increase by more than a threshold. The article proposes a differentially private release mechanism, which samples the cases in the log and adds noise to the timestamps to the extent required to achieve the above privacy guarantee. The article reports on an empirical comparison of the proposed approach against the state-of-the-art approaches using 14 real-life event logs in terms of data utility loss and computational efficiency.