Fast radio bursts (FRBs) are millisecond-duration, bright (approximately Jy) extragalactic bursts, whose production mechanism is still unclear<SUP>1</SUP>. Recently, two repeating FRBs were found to have a physically associated persistent radio source of non-thermal origin<SUP>2,3</SUP>. These two FRBs have unusually large Faraday rotation measure values<SUP>2,3</SUP>, probably tracing a dense magneto-ionic medium, consistent with synchrotron radiation originating from a nebula surrounding the FRB source<SUP>4–8</SUP>. Recent theoretical arguments predict that, if the observed Faraday rotation measure mostly arises from the persistent radio source region, there should be a simple relation between the persistent radio source luminosity and the rotation measure itself<SUP>7,9</SUP>. Here we report the detection of a third, less luminous persistent radio source associated with the repeating FRB source FRB 20201124A at a distance of 413 Mpc, substantially expanding the predicted relation into the low luminosity–low Faraday rotation measure regime (<1,000 rad m<SUP>‑2</SUP>). At lower values of the Faraday rotation measure, the expected radio luminosity falls below the limit-of-detection threshold for present-day radio telescopes. These findings support the idea that the persistent radio sources observed so far are generated by a nebula in the FRB environment and that FRBs with low Faraday rotation measure may not show a persistent radio source because of a weaker magneto-ionic medium. This is generally consistent with models invoking a young magnetar as the central engine of the FRB, in which the surrounding ionized nebula—or the interacting shock in a binary system—powers the persistent radio source.