We present year-long, near-infrared Hubble Space Telescope (HST) WFC3 observations used to search for Mira variables in NGC 1559, the host galaxy of the Type Ia supernova (SN Ia) SN 2005df. This is the first dedicated search for Miras, highly evolved low-mass stars, in an SNIa 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<400days 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 host galaxies. Using a sample of O-rich Miras discovered in NGC 4258 with HSTF160W and its maser distance, we measure a distance modulus for NGC1559 of {mu}_1559_=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 M_B_^0^=-19.27+/-0.13mag. With the Hubble diagram of SNe Ia we find H_0_=72.7+/-4.6km/s/Mpc. Combining the calibration from the NGC 4258 megamaser and the Large Magellanic Cloud detached eclipsing binaries gives a best value of H_0_=73.3+/-4.0km/s/Mpc. 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 that should reduce the error in H_0_ 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.