  A well known generalisation of Dirac's theorem states that if a graph $G$ on
$n\ge 4k$ vertices has minimum degree at least $n/2$ then $G$ contains a
$2$-factor consisting of exactly $k$ cycles. This is easily seen to be tight in
terms of the bound on the minimum degree. However, if one assumes in addition
that $G$ is Hamiltonian it has been conjectured that the bound on the minimum
degree may be relaxed. This was indeed shown to be true by S\'ark\"ozy. In
subsequent papers, the minimum degree bound has been improved, most recently to
$(2/5+\varepsilon)n$ by DeBiasio, Ferrara, and Morris. On the other hand no
lower bounds close to this are known, and all papers on this topic ask whether
the minimum degree needs to be linear. We answer this question, by showing that
the required minimum degree for large Hamiltonian graphs to have a $2$-factor
consisting of a fixed number of cycles is sublinear in $n.$
