Narrow-line Seyfert 1 galaxies have been identified by the Fermi Gamma-Ray Space Telescope as a rare class of gamma-ray emitting active galactic nuclei (AGN). The lowest-redshift candidate among them is the source 1H 0323+342. Here we present quasi-simultaneous Gemini near-infrared and Keck optical spectroscopy for it, from which we derive a black hole mass based on both the broad Balmer and Paschen emission lines. We supplement these observations with a NuSTAR X-ray spectrum taken about two years earlier, from which we constrain the black hole mass based on the short timescale spectral variability. Our multiwavelength observations suggest a black hole mass of ~2x10^7 solar masses, which agrees well with previous estimates. We build the spectral energy distribution and show that it is dominated by the thermal and reprocessed emission from the accretion disc rather than the non-thermal jet component. A detailed spectral fitting with the energy-conserving accretion disc model of Done et al. constrains the Eddington ratio to L/L_Edd ~ 0.5 for a (non-rotating) Schwarzschild black hole and to L/L_Edd ~ 1 for a Kerr black hole with dimensionless spin of a*=0.8. Higher spin values and so higher Eddington ratios are excluded, since they would strongly overpredict the observed soft X-ray flux.
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