We present deep and precise photometry (F435W, F625W, F658N) of ω Cen collected with the Advanced Camera for Surveys (ACS) on board the Hubble Space Telescope (HST). We have identified ~6500 white dwarf (WD) candidates, and the ratio of WD to main-sequence (MS) star counts is found to be at least a factor of 2 larger than the ratio of CO-core WD cooling to MS lifetimes. This discrepancy is not explained by the possible occurrence of a He-enhanced stellar population, since the MS lifetime changes by only 15% when changing from a canonical (Y=0.25) to a He-enhanced composition (Y=0.42). The presence of some He-core WDs seems able to explain the observed star counts. The fraction of He WDs required ranges from 10% to 80% depending on their mean mass, and it is at least 5 times larger than for field WDs. The comparison in the color-magnitude diagram between theory and observations also supports the presence of He WDs. Empirical evidence indicates that He WDs have been detected in stellar systems hosting a large sample of extreme horizontal branch stars, thus suggesting that a fraction of red giants might avoid the He-core flash. Based on observations collected with the Advanced Camera for Surveys on board the Hubble Space Telescope.