Let $\mathrm{SO}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $μ$. Resolving a problem by Breuillard and Green, we show that if $A \subseteq \mathrm{SO}(3,\mathbb{R})$ is an open subset with sufficiently small measure, then $$ μ(A^2) > 3.99 μ(A).$$ We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in $\mathrm{SO}(3,\mathbb{R})$.