This paper studies the relaying strategies and the approximate capacity of
the classic three-node Gaussian relay channel, but where the noises at the
relay and at the destination are correlated. It is shown that the capacity of
such a relay channel can be achieved to within a constant gap of $\hf \log_2 3
=0.7925$ bits using a modified version of the noisy network coding strategy,
where the quantization level at the relay is set in a correlation dependent
way. As a corollary, this result establishes that the conventional
compress-and-forward scheme also achieves to within a constant gap to the
capacity. In contrast, the decode-and-forward and the single-tap
amplify-and-forward relaying strategies can have an infinite gap to capacity in
the regime where the noises at the relay and at the destination are highly
correlated, and the gain of the relay-to-destination link goes to infinity.