  We study a Markovian model for the random fragmentation of an object. At each
time, the state consists of a collection of blocks. Each block waits an
exponential amount of time with parameter given by its size to some power
$\alpha$, independently of the other blocks. Every block then splits randomly
into sub-blocks whose relative sizes are distributed according to the so-called
dislocation measure. We focus here on the case where $\alpha<0$. In this case,
small blocks split intensively, and so the whole state is reduced to "dust" in
a finite time, almost surely (we call this the extinction time). In this paper,
we investigate how the fragmentation process behaves as it approaches its
extinction time. In particular, we prove a scaling limit for the block sizes
which, as a direct consequence, gives us an expression for an invariant measure
for the fragmentation process. In an earlier paper [Ann. Inst. Henri Poincar\'e
Probab. Stat. 46 (2010) 338-368], we considered the same problem for another
family of fragmentation processes, the so-called stable fragmentations. The
results here are similar, but we emphasize that the methods used to prove them
are different. Our approach in the present paper is based on Markov renewal
theory and involves a somewhat unusual "spine" decomposition for the
fragmentation, which may be of independent interest.
