Exomoons represent a crucial missing puzzle piece in our efforts to understand extrasolar planetary systems. To address this deficiency, we here describe an exomoon survey of 70 cool, giant transiting exoplanet candidates found by Kepler. We identify only one exhibiting a moon-like signal that passes a battery of vetting tests: Kepler-1708b. We show that Kepler-1708b is a statistically validated Jupiter-sized planet orbiting a Sun-like quiescent star at 1.6au. The signal of the exomoon candidate, Kepler-1708 b-i, is a 4.8{sigma} effect and is persistent across different instrumental detrending methods, with a 1% false-positive probability via injection-recovery. Kepler-1708b-i is ~2.6 Earth radii and is located in an approximately coplanar orbit at ~12 planetary radii from its ~1.6au Jupiter-sized host. Future observations will be necessary to validate or reject the candidate.