  We study the distribution of finite clusters in slightly supercritical ($p
\downarrow p_c$) Bernoulli bond percolation on transitive nonamenable graphs,
proving in particular that if $G$ is a transitive nonamenable graph satisfying
the $L^2$ boundedness condition ($p_c<p_{2\to 2}$) and $K$ denotes the cluster
of the origin then there exists $\delta>0$ such that $$ \mathbf{P}_p(n \leq |K|
< \infty) \asymp n^{-1/2} \exp\left[ -\Theta \Bigl( |p-p_c|^2 n\Bigr) \right]
$$ and \[ \mathbf{P}_p(r \leq \operatorname{Rad}(K) < \infty) \asymp r^{-1}
\exp\left[ -\Theta \Bigl( |p-p_c| r\Bigr) \right] \] for every $p\in
(p_c-\delta,p_c+\delta)$ and $n,r\geq 1$, where all implicit constants depend
only on $G$. We deduce in particular that the critical exponents $\gamma'$ and
$\Delta'$ describing the rate of growth of the moments of a finite cluster as
$p \downarrow p_c$ take their mean-field values of $1$ and $2$ respectively.
  These results apply in particular to Cayley graphs of nonelementary
hyperbolic groups, to products with trees, and to transitive graphs of spectral
radius $\rho<1/2$. In particular, every finitely generated nonamenable group
has a Cayley graph to which these results apply. They are new for graphs that
are not trees. The corresponding facts are yet to be understood on
$\mathbb{Z}^d$ even for $d$ very large. In a second paper in this series, we
will apply these results to study the geometric and spectral properties of
infinite slightly supercritical clusters in the same setting.
