  Let $X$ be a smooth complex projective curve of genus $g\geq 3$. Let
$\mathbf{M}_2$ be the moduli space of semistable rank $2$ Higgs bundles with
trivial determinant over $X$. We construct a desingularization $\mathbf{S}$ of
$\mathbf{M}_2$ as a closed subvariety of a moduli space. We prove that
$\mathbf{S}$ is a nonsingular variety containing the stable locus of
$\mathbf{M}_2$ as an open dense subvariety. On the other hand, there is another
desingularization $\mathbf{K}$ of $\mathbf{M}_2$ obtained from Kirwan's
algorithm. We show that $\mathbf{S}$ can be obtained after two blow-downs of
$\mathbf{K}$.
