  A regenerative random composition of integer $n$ is constructed by allocating
$n$ standard exponential points over a countable number of intervals,
comprising the complement of the closed range of a subordinator $S$. Assuming
that the L\'{e}vy measure of $S$ is infinite and regularly varying at zero of
index $-\alpha$, $\alpha\in(0,\,1)$, we find an explicit threshold $r=r(n)$,
such that the number $K_{n,\,r(n)}$ of blocks of size $r(n)$ converges in
distribution without any normalization to a mixed Poisson distribution. The
sequence $(r(n))$ turns out to be regularly varying with index
$\alpha/(\alpha+1)$ and the mixing distribution is that of the exponential
functional of $S$. The result is derived as a consequence of a general Poisson
limit theorem for an infinite occupancy scheme with power-like decay of the
frequencies. We also discuss asymptotic behavior of $K_{n,\,w(n)}$ in cases
when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement
previously known strong laws of large numbers for $K_{n,\,r}$ in case of a
fixed $r\in\mathbb{N}$. As a key tool we employ new Abelian theorems for
Laplace--Stiletjes transforms of regularly varying functions with the indexes
of regular variation diverging to infinity.
