  The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the
minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair
of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a
subgraph. We study $\mathrm{ex}_2(n, F)$ when $F=K_4^-$, the $3$-graph on $4$
vertices with $3$ edges. Using flag algebra techniques, we prove that if $n$ is
sufficiently large then $\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4$.
  This settles in the affirmative a conjecture of Nagle from 1999. In addition,
we obtain a stability result: for every near-extremal configuration $G$, there
is a quasirandom tournament $T$ on the same vertex set such that $G$ is close
in the edit distance to the $3$-graph $C(T)$ whose edges are the cyclically
oriented triangles from $T$. For infinitely many values of $n$, we are further
able to determine $\mathrm{ex}_2(n, K_4^-)$ exactly and to show that
tournament-based constructions $C(T)$ are extremal for those values of $n$.
