  We construct a left adjoint $\mathcal{H}^\text{st}\colon \mathbf{Ex}_{\infty}
\rightarrow \mathbf{St}_{\infty}$ to the inclusion $\mathbf{St}_{\infty}
\hookrightarrow \mathbf{Ex}_{\infty}$ of the $\infty$-category of stable
$\infty$-categories into the $\infty$-category of exact $\infty$-categories,
which we call the stable hull. For every exact $\infty$-category $\mathcal{E}$,
the unit functor $\mathcal{E} \rightarrow \mathcal{H}^\text{st}(\mathcal{E})$
is fully faithful and preserves and reflects exact sequences. This provides an
$\infty$-categorical variant of the Gabriel-Quillen embedding for ordinary
exact categories. If $\mathcal{E}$ is an ordinary exact category, the stable
hull $\mathcal{H}^\text{st}(\mathcal{E})$ is equivalent to the bounded derived
$\infty$-category of $\mathcal{E}$.
