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The detection of several radio-loud narrow-line Seyfert 1 (NLS1) galaxies by the Fermi Gamma-Ray Space Telescope hints at the existence of a rare, new class of gamma-ray emitting active galactic nuclei with low black hole masses. Like flat spectrum radio quasars (FSRQs), their gamma-ray emission is thought to be produced via the external Compton mechanism whereby relativistic jet electrons upscatter a photon field external to the jet, e.g. from the accretion disc, broad line region (BLR) and dusty torus, to higher energies. Here we study the origin of the gamma-ray emission in the lowest-redshift candidate among the currently-known gamma-ray emitting NLS1s, 1H 0323+342, and take a new approach. We observationally constrain the external photon field using quasi-simultaneous near-IR, optical and X-ray spectroscopy. Applying a one-zone leptonic jet model, we simulate the range of jet parameters for which this photon field, when Compton scattered to higher energies, can explain the gamma-ray emission. We find that the site of the gamma-ray emission lies well within the BLR and that the seed photons mainly originate from the accretion disc. The jet power that we determine, $1.0 \times 10^{45}$ erg s$^{-1}$, is approximately half the accretion disc luminosity. We show that this object is not simply a low-mass FSRQ, its jet is intrinsically less powerful than predicted by scaling a typical FSRQ jet by black hole mass and accretion rate. That gamma-ray emitting NLS1s appear to host underpowered jets may go some way to explaining why so few have been detected to date., Comment: 22 pages, 8 figures and 8 tables. Accepted for publication in MNRAS