  CUORE, an array of 988 TeO$_2$ bolometers, is about to be one of the most
sensitive experiments searching for neutrinoless double-beta decay. Its
sensitivity could be further improved by removing the background from $\alpha$
radioactivity. A few years ago it has been pointed out that the signal from
$\beta$s can be tagged by detecting the emitted Cherenkov light, which is not
produced by $\alpha$s. In this paper we confirm this possibility. For the first
time we measured the Cherenkov light emitted by a CUORE crystal, and found it
to be 100 eV at the $Q$-value of the decay. To completely reject the $\alpha$
background, we compute that one needs light detectors with baseline noise below
20 eV RMS, a value which is 3-4 times smaller than the average noise of the
bolometric light detectors we are using. We point out that an improved light
detector technology must be developed to obtain TeO$_2$ bolometric experiments
able to probe the inverted hierarchy of neutrino masses.
