We present optical, infrared (IR) and radio observations of the powerful Fanaroff-Riley type II (FR II) radio source PKS 0347+05 (z = 0.3390), and demonstrate that it is a rare example of a radio-loud/radio-quiet double active galactic nucleus (AGN) system, comprising a weak-line radio galaxy (WLRG) separated by 25kpc (in projection) from a Seyfert 1 nucleus at the same redshift. Our deep Gemini optical images show a highly disturbed morphology, with a warped dust lane crossing through the halo and nuclear regions of the radio galaxy host, tidal tails and a bridge connecting the radio galaxy to the Seyfert 1 nucleus. Spectral synthesis modelling of our Gemini optical spectrum of the radio galaxy shows evidence for a reddened young stellar population of age ≤100Myr. Further evidence for recent star formation activity in this source is provided by the detection of strong polycyclic aromatic hydrocarbon features in mid-IR Spitzer/IRS spectra. Together, these observations support a model in which both AGN have been triggered simultaneously in a major galaxy merger. However, despite the presence of a powerful FR II radio source, and the apparently plentiful supply of fuel provided by the merger, the nucleus of the radio galaxy shows only weak, low-ionization emission-line activity. We speculate that the fuel supply to nuclear regions of the radio galaxy has recently switched off (within the last ∼10 6yr), but the information about the resulting decrease in nuclear AGN activity has yet to reach the extended lobes and hotspots of the FR II radio source. Based on this scenario, we derive a lower limit on the typical lifetimes of powerful, intermediate-redshift FR II radio sources of τ FR II ≳5×106yr. Overall, our observations emphasize that the fuelling of AGN activity in major galaxy mergers is likely to be highly intermittent.