We report the results of an HI blind survey of 80deg^2 of the Virgo cluster, based on the 08deg < delta < 16 deg strip of ALFALFA, the Arecibo Legacy Fast ALFA Survey. 187 HI sources of high significance are found providing a complete census of HI sources in this region of the Virgo cluster (-1000<cz<3000 km/s) with M_HI > 10^7.5-8 M_sun. 156/187 (83%) sources are identified with optical galaxies from the Virgo Cluster Catalogue (Binggeli et al. 1985), all but 8 with late-type galaxies. Ten sources are not associated with optical galaxies and were found to correspond to tidally-disrupted systems (see Kent et al. 2007 and Haynes, Giovanelli and Kent 2007). The remaining 21 (11%) are associated with galaxies that are not listed in the Virgo Cluster Catalogue. For all sources with an optical counterpart in the Sloan Digital Sky Survey, we analyzed i-band SDSS plates to measure optical structural parameters. We find that in the Virgo cluster: i) HI inhabits galaxies that are structurally similar to ordinary late-type galaxies; ii) their HI content can be predicted from their optical luminosity; iii) low surface brightness galaxies have low optical luminosity and contain small quantities of neutral hydrogen; iv) low surface brightness, massive Malin1 type galaxies are comfortably rare objects (less than 0.5%); v) there are no dark-galaxies with HI masses M_HI > 10^7.5-8 M_sun; vi) less than 1% of early-type galaxies contain neutral hydrogen with M_HI>10^7.5-8 M_sun (di Serego Alighieri et al. 2007).