We report the confirmation of an old, metal-poor globular cluster (GC) in the nearby dwarf irregular galaxy Sextans A, the first GC known in this galaxy. The cluster, which we designate as Sextans A-GC1, lies some 4.4 arcmin (˜1.8 kpc) to the SW of the galaxy centre and clearly resolves into stars in sub-arcsecond seeing ground-based imaging. We measure an integrated magnitude V = 18.04, corresponding to an absolute magnitude, MV,0 = -7.85. This gives an inferred mass M ˜ 1.6 × 105 M⊙, assuming a Kroupa IMF. An integrated spectrum of Sextans A-GC1 reveals a heliocentric radial velocity vhelio = 305 ± 15 km s-1, consistent with the systemic velocity of Sextans A. The location of candidate red giant branch stars in the cluster, and stellar population analyses of the cluster's integrated optical spectrum, suggests a metallicity [Fe/H] ˜ -2.4, and an age ˜9 Gyr. We measure a half-light radius, Rh = 7.6 ± 0.2 pc. Normalizing to the galaxy integrated magnitude, we obtain a V-band specific frequency, SN = 2.1. We compile a sample of 1928 GCs in 28 galaxies with spectroscopic metallicities and find that the low metallicity of Sextans A-GC1 is close to a `metallicity floor' at [Fe/H] ˜ -2.5 seen in these GC systems that include the Milky Way, M31, M87, and the Large Magellanic Cloud. This metallicity floor appears to hold across 6 dex in host galaxy stellar mass and is seen in galaxies with and without accreted GC subpopulations.