  We use data from the ESA Gaia mission Early Data Release 3 (EDR3) to measure
the trigonometric parallax of $\omega$ Cen, the first high precision parallax
measurement for the most massive globular cluster in the Milky Way. We use a
combination of positional and high quality proper motion data from EDR3 to
identify over 100,000 cluster members, of which 67,000 are in the magnitude and
color range where EDR3 parallaxes are best calibrated. We find the estimated
parallax to be robust, demonstrating good control of systematics within the
color-magnitude diagram of the cluster. We find a parallax for the cluster of
$0.191\pm0.001$ (statistical) $\pm0.004$ (systematic) mas (2.2\% total
uncertainty) corresponding to a distance of $5.24\pm0.11$ kpc. The parallax of
$\omega$ Cen provides a unique opportunity to directly and geometrically
calibrate the luminosity of the Tip of the Red Giant Branch (TRGB) because it
is the only cluster with sufficient mass to provide enough red giant stars,
more than 100 one magnitude below the tip, for a precise, model-free
measurement of the tip. Combined with the pre-existing and most widely-used
measurements of the tip and foreground Milky Way extinction, we find
$M_{I,TRGB}=-3.97\pm0.06$ mag for the $I$-band luminosity of the blue edge.
Using the TRGB luminosity calibrated from the Gaia EDR3 parallax of $\omega$
Cen to calibrate the luminosity of SNIa results in a value for the Hubble
constant of $H_0=72.1\pm2.0$ km s$^{-1}$ Mpc$^{-1}$. We make the data for the
stars in $\omega$ Cen available electronically and encourage independent
analyses of the results presented here.
