= Solution
The log likelihood, up to constants, is
$$
-\frac n2\log\sigma^2
-\frac1{2\sigma^2}(Y-X\beta)^T\Sigma_0^{-1}(Y-X\beta).
$$
Differentiation gives the <generalized least squares>[generalized least squares estimator]
$$
\boxed{
\widehat\beta=(X^T\Sigma_0^{-1}X)^{-1}X^T\Sigma_0^{-1}Y},
$$
and maximizing over the scale gives
$$
\boxed{
\widehat\sigma^2
=\frac1n(Y-X\widehat\beta)^T\Sigma_0^{-1}(Y-X\widehat\beta)}.
$$
The divisor $n$ is appropriate for maximum likelihood rather than unbiased estimation.
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