= Solution
Write $\widehat\beta=AY$, where
$$
A=(X^T\Sigma_0^{-1}X)^{-1}X^T\Sigma_0^{-1},
\qquad AX=I.
$$
Any other linear unbiased estimator is $CY$ with $CX=I$, so $C=A+D$ and $DX=0$. The cross covariance vanishes because
$$
A\Sigma_0D^T
=(X^T\Sigma_0^{-1}X)^{-1}X^TD^T=0.
$$
Therefore
$$
\operatorname{Cov}(CY)-\operatorname{Cov}(AY)
=\sigma^2D\Sigma_0D^T\succeq0.
$$
By the <Gauss-Markov theorem>,
$$
\boxed{\widehat\beta\text{ is the best linear unbiased estimator of }\beta}.
$$
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