  The C_\ell-free process starts with the empty graph on n vertices and adds
edges chosen uniformly at random, one at a time, subject to the condition that
no copy of C_\ell is created. For every $\ell \geq 4$ we show that, with high
probability as $n \to \infty$, the maximum degree is $O((n \log
n)^{1/(\ell-1)})$, which confirms a conjecture of Bohman and Keevash and
improves on bounds of Osthus and Taraz. Combined with previous results this
implies that the C_\ell-free process typically terminates with
$\Theta(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)})$ edges, which answers a
question of Erd\H{o}s, Suen and Winkler. This is the first result that
determines the final number of edges of the more general H-free process for a
non-trivial \emph{class} of graphs H. We also verify a conjecture of Osthus and
Taraz concerning the average degree, and obtain a new lower bound on the
independence number. Our proof combines the differential equation method with a
tool that might be of independent interest: we establish a rigorous way to
`transfer' certain decreasing properties from the binomial random graph to the
H-free process.
