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 ± 0.023 RJ) exoplanet Kepler-1514 b with a 218-day orbital period as 5.28 ± 0.22MJ. The bulk density of this cool (~390 K) giant planet is -4.82+0.26-0.25g 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 as one that is amenable to continued study.