  We present year-long, near-infrared Hubble Space Telescope WFC3 observations
used to search for Mira variables in NGC 1559, the host galaxy of the Type Ia
supernova (SN Ia) 2005df. This is the first dedicated search for Miras,
highly-evolved low-mass stars, in a SN Ia host and subsequently the first
calibration of the SN Ia luminosity using Miras in a role historically played
by Cepheids. We identify a sample of 115 O-rich Miras with P < 400 days based
on their light curve properties. We find that the scatter in the Mira
Period-Luminosity Relation (PLR) is comparable to Cepheid PLRs seen in SN Ia
supernova host galaxies. Using a sample of O-rich Miras discovered in NGC 4258
with HST F160W and its maser distance, we measure a distance modulus for NGC
1559 of mu1559 = 31.41 +/- 0.050 (statistical) +/- 0.060 (systematic) mag.
Based on the light curve of the normal, well-observed, low-reddening SN 2005df,
we obtain a measurement of the fiducial SN Ia absolute magnitude of MB0 =
-19.27 +/- 0.13 mag. With the Hubble diagram of SNe Ia we find H0 = 72.7 +/-
4.6 kms-1 Mpc-1. Combining the calibration from the NGC 4258 megamaser and the
Large Magellanic Cloud detached eclipsing binaries gives a best value of H0 =
73.3 +/- 4.0 km s-1 Mpc-1. This result is within 1-sigma of the Hubble constant
derived using Cepheids and multiple calibrating SNe Ia. This is the first of
four expected calibrations of the SN Ia luminosity from Miras which should
reduce the error in H0 via Miras to ~3%. In light of the present Hubble tension
and JWST, Miras have utility in the extragalactic distance scale to check
Cepheid distances or calibrate nearby SNe in early-type host galaxies that
would be unlikely targets for Cepheid searches.
