  Let $k\geq 3$. Given a $k$-uniform hypergraph $H$, the minimum codegree
$\delta(H)$ is the largest $d\in\mathbb{N}$ such that every $(k-1)$-set of
$V(H)$ is contained in at least $d$ edges. Given a $k$-uniform hypergraph $F$,
the codegree Tur\'an density $\gamma(F)$ of $F$ is the smallest $\gamma \in
[0,1]$ such that every $k$-uniform hypergraph on $n$ vertices with
$\delta(H)\geq (\gamma + o(1))n$ contains a copy of $F$. Similarly as other
variants of the hypergraph Tur\'an problem, determining the codegree Tur\'an
density of a hypergraph is in general notoriously difficult and only few
results are known.
  In this work, we show that for every $\varepsilon>0$, there is a $k$-uniform
hypergraph $F$ with $0<\gamma(F)<\varepsilon$. This is in contrast to the
classical Tur\'an density, which cannot take any value in the interval
$(0,k!/k^k)$ due to a fundamental result by Erd\H{o}s.
