  The Gauss-Markov source produces $U_i = aU_{i-1} + Z_i$ for $i\geq 1$, where
$U_0 = 0$, $|a|<1$ and $Z_i\sim\mathcal{N}(0, \sigma^2)$ are i.i.d. Gaussian
random variables. We consider lossy compression of a block of $n$ samples of
the Gauss-Markov source under squared error distortion. We obtain the Gaussian
approximation for the Gauss-Markov source with excess-distortion criterion for
any distortion $d>0$, and we show that the dispersion has a reverse
waterfilling representation. This is the \emph{first} finite blocklength result
for lossy compression of \emph{sources with memory}. We prove that the finite
blocklength rate-distortion function $R(n,d,\epsilon)$ approaches the
rate-distortion function $\mathbb{R}(d)$ as $R(n,d,\epsilon) = \mathbb{R}(d) +
\sqrt{\frac{V(d)}{n}}Q^{-1}(\epsilon) + o\left(\frac{1}{\sqrt{n}}\right)$,
where $V(d)$ is the dispersion, $\epsilon \in (0,1)$ is the excess-distortion
probability, and $Q^{-1}$ is the inverse of the $Q$-function. We give a reverse
waterfilling integral representation for the dispersion $V(d)$, which parallels
that of the rate-distortion functions for Gaussian processes. Remarkably, for
all $0 < d\leq \frac{\sigma^2}{(1+|a|)^2}$, $R(n,d,\epsilon)$ of the
Gauss-Markov source coincides with that of $Z_k$, the i.i.d. Gaussian noise
driving the process, up to the second-order term. Among novel technical tools
developed in this paper is a sharp approximation of the eigenvalues of the
covariance matrix of $n$ samples of the Gauss-Markov source, and a construction
of a typical set using the maximum likelihood estimate of the parameter $a$
based on $n$ observations.
