  Given a rational pointed $n$-dimensional cone $C$, we study the integer
Carath\'{e}odory rank $\operatorname{CR}(C)$ and its asymptotic form
$\operatorname{CR^{\rm a}}(C)$, where we consider ``most'' integer vectors in
the cone. The main result significantly improves the previously known upper
bound for $\operatorname{CR^{\rm a}}(C)$. We also study bounds on
$\operatorname{CR}(C)$ in terms of $\Delta$, the maximal absolute $n\times n$
minor of the matrix given in an integral polyhedral representation of $C$. If
$\Delta\in\lbrace 1,2\rbrace$, we show $\operatorname{CR}(C) = n$, and prove
upper bounds for simplicial cones, improving the best known upper bound on
$\operatorname{CR}(C)$ for $\Delta\leq n$.
