We present the localization and host galaxy of FRB 20190208A, a repeating source of fast radio bursts (FRBs) discovered using CHIME/FRB. As part of the Pinpointing REpeating ChIme Sources with EVN dishes repeater localization program on the European VLBI Network (EVN), we monitored FRB 20190208A for 65.6 hr at ∼1.4 GHz and detected a single burst, which led to its very long baseline interferometry localization with 260 mas uncertainty (2σ). Follow-up optical observations with the MMT Observatory (i ≳ 25.7 mag (AB)) found no visible host at the FRB position. Subsequent deeper observations with the Gran Telescopio Canarias, however, revealed an extremely faint galaxy (r = 27.32 ± 0.16 mag), very likely (99.95%) associated with FRB 20190208A. Given the dispersion measure of the FRB (∼580 pc cm<SUP>‑3</SUP>), even the most conservative redshift estimate (<inline-formula> </inline-formula>) implies that this is the lowest-luminosity FRB host to date (≲10<SUP>8</SUP> L <SUB>⊙</SUB>), even less luminous than the dwarf host of FRB 20121102A. We investigate how localization precision and the depth of optical imaging affect host association and discuss the implications of such a low-luminosity dwarf galaxy. Unlike the other repeaters with low-luminosity hosts, FRB 20190208A has a modest Faraday rotation measure of a few tens of rad m<SUP>‑2</SUP>, and EVN plus Very Large Array observations reveal no associated compact persistent radio source. We also monitored FRB 20190208A for 40.4 hr over 2 yr as part of the Extragalactic Coherent Light from Astrophysical Transients repeating FRB monitoring campaign on the Nançay Radio Telescope and detected one burst. Our results demonstrate that, in some cases, the robust association of an FRB with a host galaxy will require both high localization precision and deep optical follow-up.