  We present an analysis of the transit timing variations (TTVs) in the
multi-transiting planetary system around Kepler-51 (KOI-620). This system
consists of two confirmed transiting planets, Kepler-51b ($P_{\rm b} =
45.2\,\mathrm{days}$) and Kepler-51c ($P_{\rm c} = 85.3\,\mathrm{days}$), and
one transiting planet candidate KOI-620.02 ($P_{02} = 130.2\,\mathrm{days}$),
which lie close to a $1:2:3$ resonance chain. Our analysis shows that their
TTVs are consistently explained by the three-planet model, and constrains their
masses as $M_{\rm b} = 2.1_{-0.8}^{+1.5} M_{\oplus}$ (Kepler-51b), $M_{\rm c} =
4.0 \pm 0.4 M_{\oplus}$ (Kepler-51c), and $M_{02} = 7.6 \pm 1.1 M_{\oplus}$
(KOI-620.02), thus confirming KOI-620.02 as a planet in this system. The masses
inferred from the TTVs are rather small compared to the planetary radii based
on the stellar density and planet-to-star radius ratios determined from the
transit light curves. Combining these estimates, we find that all three planets
in this system have densities among the lowest determined, $\rho_p \lesssim
0.05\,{\rm g\,cm^{-3}}$. With this feature, the Kepler-51 system serves as
another example of low-density compact multi-transiting planetary systems. We
also identify a curious feature in the archived Kepler light curve during the
double transit of Kepler-51b and KOI-620.02, which could be explained by their
overlapping on the stellar disk (a planet-planet eclipse). If this is really
the case, the sky-plane inclination of KOI-620.02's orbit relative to that of
Kepler-51b is given by $\Delta \Omega = -25.3_{-6.8}^{+6.2}\mathrm{deg}$,
implying significant misalignment of their orbital planes. This interpretation,
however, seems unlikely because such an event that is consistent with all of
the observations is found to be exceedingly rare.
