We present an earthquake source inversion using a single time series produced by integrated fiber-optic sensing in a phase noise cancellation (PNC) system used for frequency metrology. Operating on a 123 km long fiber between Bern and Basel (Switzerland), the PNC system recorded the Mw3.9 Mulhouse earthquake that occurred on 10 September 2022 around 10 km north-west of the northern fiber end. A generalised least-squares inversion in the 4 - 13 s period band constrains the components of a double-couple moment tensor with an uncertainty that corresponds to around 0.2 moment magnitude units, nearly independent of prior information. Uncertainties for hypocenter location and original time are more variable, ranging between 4 - 20 km and 0.1 - 1 s, respectively, depending on whether injected prior information is realistic or almost absent. This work is a proof of concept that quantifies the resolvability of earthquake source properties under specific conditions using a single-channel stand-alone integrated (non-distributed) fiber-optic measurement. It thereby constitutes a step towards the integration of long-range phase-transmission fiber-optic sensors into existing seismic networks in order to fill significant seismic data gaps, especially in the oceans.