Damped Lyman-α absorbers (DLAs) as a class of quasi-stellar object (QSO) absorption-line systems are currently our most important source of detailed information on the cosmic chemical evolution of galaxies. However, the degree to which this information is biased by dust remains to be understood. One strategy is to specifically search for QSOs that are reddened by metal-rich and dusty foreground absorbers. In this Letter, we present the discovery of a z = 2.60 QSO that is strongly reddened by dust in an intervening extremely strong DLA at z = 2.226. This QSO was identified through a novel selection that combined the astrometric measurements from ESA's Gaia satellite with extisting optical and near- to mid-infrared photometry. We infer a total neutral atomic-hydrogen column density of log N(H I) = 21.95 ± 0.15 and a lower limit on the gas-phase metallicity of [Zn/H] > -0.96. This DLA is also remarkable in that it exhibits shielded neutral gas that is visible in C I, and it shows tentative detections of CO molecular bands. The spectral energy distribution of the QSO is well accounted for by a spectral energy distribution that is normal for a QSO and is reddened by dust from a DLA with 10% solar metallicity, a dust extinction of AV = 0.82 ± 0.02 mag, and an extinction curve that resembles that of the Large Magellanic Cloud, including the characteristic 2175 Å extinction feature. Such QSO absorption-line systems have been shown to be very rare in previous surveys, which have mostly revealed sight-lines with low extinction. The present case therefore suggests that previous samples have under-represented the fraction of dusty absorbers. Building a complete sample of such systems is required to assess the significance of this effect.