We present a deep VLA search for HI emission from the low-luminosity Virgo Cluster elliptical galaxy NGC 4476, which contains 1.1 x 10^8 M_sun of molecular gas in an undisturbed disk in regular rotation. No HI was detected. The rms noise in the final image corresponds to a 3 sigma column density sensitivity of 1.2 x 10^20 cm^{-2} at the position of NGC 4476, averaged over the 4 kpc beam. The total HI mass is less than 1.5 x 10^7 M_sun. If we compare our HI upper limit to the H_2 content, we find that NGC 4476 is extremely deficient in HI compared to other galaxies detected in these two species. The H_2/HI mass ratio for NGC 4476 is > 7, whereas typical H_2/HI ratios for elliptical galaxies detected in both HI and H_2 are <~2. Based on this extreme HI deficiency and the intra-cluster medium (ICM) density at the projected distance from M87 we argue that either NGC 4476 has undergone ram-pressure stripping while traveling through the Virgo cluster core or its average molecular gas density is larger and its interstellar UV field is smaller than in typical spiral galaxies. NGC 4476 is located 12 in projection from M87, which causes extreme continuum confusion problems. We also discuss in detail the techniques used for continuum subtraction. The spectral dynamic range of our final image is 50,000 to 1.
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