  Despite the severe bias of the transit method of exoplanet discovery toward
short orbital periods, a modest sample of transiting exoplanets with orbital
periods greater than 100 days is known. Long-term radial velocity (RV) surveys
are pivotal to confirming these signals and generating a set of planetary
masses and densities for planets receiving moderate to low irradiation from
their host stars. Here, we conduct RV observations of Kepler-1514 from the Keck
I telescope using the High Resolution Echelle Spectrometer. From these data, we
measure the mass of the statistically validated giant ($1.108\pm0.023$ $R_{\rm
J}$) exoplanet Kepler-1514 b with a 218 day orbital period as $5.28\pm0.22$
$M_{\rm J}$. The bulk density of this cool ($\sim$390 K) giant planet is
$4.82^{+0.26}_{-0.25}$ g cm$^{-3}$, consistent with a core supported by
electron degeneracy pressure. We also infer an orbital eccentricity of
$0.401^{+0.013}_{-0.014}$ from the RV and transit observations, which is
consistent with planet-planet scattering and disk cavity migration models. The
Kepler-1514 system contains an Earth-size, Kepler Object of Interest on a 10.5
day orbit that we statistically validate against false positive scenarios,
including those involving a neighboring star. The combination of the brightness
($V$=11.8) of the host star and the long period, low irradiation, and high
density of Kepler-1514 b places this system among a rare group of known
exoplanetary systems and one that is amenable to continued study.
