  The purpose of this article is to investigate Bach-flat critical metrics of
the volume functional on a compact manifold $M$ with boundary $\partial M.$
Here, we prove that a Bach-flat critical metric of the volume functional on a
simply connected 4-dimensional manifold with boundary isometric to a standard
sphere must be isometric to a geodesic ball in a simply connected space form
$\Bbb{R}^{4},$ $\Bbb{H}^{4}$ or $\Bbb{S}^{4}$. Moreover, we show that in
dimension three the result even is true replacing the Bach-flat condition by
the weaker assumption that $M$ has divergence-free Bach tensor.
