  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.
