  We report the continuous and partially nondestructive measurement of optical
photons. For a weak light pulse traveling through a slow-light optical medium
(signal), the associated atomic-excitation component is detected by another
light beam (probe) with the aid of an optical cavity. We observe strong
correlations of $g^{(2)}_{sp}=4.4(5)$ between the transmitted signal and probe
photons. The observed (intrinsic) conditional nondestructive quantum efficiency
ranges between 13% and 1% (65% and 5%) for a signal transmission range of 2% to
35%, at a typical time resolution of 2.5 $\mu$s. The maximal observed
(intrinsic) device nondestructive quantum efficiency, defined as the product of
the conditional nondestructive quantum efficiency and the signal transmission,
is 0.5% (2.4%). The normalized cross-correlation function violates the
Cauchy-Schwarz inequality, confirming the non-classical character of the
correlations.
