  We report integrated orbital fits for the inner regular moons of Neptune
based on the most complete astrometric data set to date, with observations from
Earth-based telescopes, Voyager 2, and the Hubble Space Telescope covering
1981-2016. We summarize the results in terms of state vectors, mean orbital
elements, and orbital uncertainties. The estimated masses of the two innermost
moons, Naiad and Thalassa, are $GM_{Naiad}$= 0.0080 $\pm$ 0.0043 $km^3 s^{-2}$
and $GM_{Thalassa}$=0.0236 $\pm$ 0.0064 $km^3 s^{-2}$, corresponding to
densities of 0.80 $\pm$ 0.48 $g cm^{-3}$ and 1.23 $\pm$ 0.43 $g cm^{-3}$,
respectively. Our analysis shows that Naiad and Thalassa are locked in an
unusual type of orbital resonance. The resonant argument 73
$\dot{\lambda}_{Thalassa}$-69 $\dot{\lambda}_{Naiad}$-4 $\dot{\Omega}_{Naiad}$
$\approx$ 0 librates around 180 deg with an average amplitude of ~66 deg and a
period of ~1.9 years for the nominal set of masses. This is the first
fourth-order resonance discovered between the moons of the outer planets. More
high precision astrometry is needed to better constrain the masses of Naiad and
Thalassa, and consequently, the amplitude and the period of libration. We also
report on a 13:11 near-resonance of Hippocamp and Proteus, which may lead to a
mass estimate of Proteus provided that there are future observations of
Hippocamp. Our fit yielded a value for Neptune's oblateness coefficient of
$J_2$=3409.1$\pm$2.9 $\times 10^{-6}$.
