  Given hypergraphs $F$ and $H$, an $F$-factor in $H$ is a set of
vertex-disjoint copies of $F$ which cover all the vertices in $H$. Let $K^- _4$
denote the $3$-uniform hypergraph with $4$ vertices and $3$ edges. We show that
for sufficiently large $n\in 4 \mathbb N$, every $3$-uniform hypergraph $H$ on
$n$ vertices with minimum codegree at least $n/2-1$ contains a $K^- _4$-factor.
Our bound on the minimum codegree here is best-possible. It resolves a
conjecture of Lo and Markstr\"om for large hypergraphs, who earlier proved an
asymptotically exact version of this result. Our proof makes use of the
absorbing method as well as a result of Keevash and Mycroft concerning almost
perfect matchings in hypergraphs.
