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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.