We present the discovery of a reflection-dominated active galactic nucleus (AGN) in the early-type radio-quiet galaxy ESO 565-G019 with Suzaku and Swift/Burst Alert Telescope. The source X-ray spectrum below 10 keV is characteristic of other Compton-thick (CT) AGNs, clearly showing an inverted continuum and prodigious fluorescence iron emission above similar to 3 keV. A Compton shoulder to the neutral Fe K alpha line also appears to be present. There is evidence for long-term hard X-ray flux variability that we associate with changes in the intrinsic AGN power law. More of such reflection-dominated AGNs should be uncovered in the near future with the increased sensitivity of ongoing and new hard X-ray surveys. ESO 565-G019 is hosted in an early-type galaxy whose morphology has been variously classified as either type E or type S0. Only about 20 bona fide CT-AGNs have been identified in the local universe so far, and all exist in host galaxies with late Hubble types (S0 or later). CT columns of nuclear obscuring gas are uncommon in early-type galaxies in the local universe, so confirmation of the exact morphological class of ESO 565-G019 is important. Infrared photometry also shows the presence of large quantities of cool dust in the host, indicative of significant ongoing star formation. ESO 565-G019 may be the first identified local example of minor-merger-driven CT-AGN growth in an early-type host, or may be the result of interaction with its neighboring galaxy ESO 565-G018 in a wide pair.