  We extend Bollobas' classical result on the chromatic number of a binomial
random graph to the exchangeable random graph model $\mathcal{G}(n,W)$ defined
by a graphon $W:[0,1]^2 \rightarrow [0,1]$, which is a symmetric measurable
function. In the case when $W$ can be approximated by block graphons in
$\mathcal{L}^{\infty}$-norm, we show that asymptotically optimal value of the
number of colours required for $\mathcal{G}(n,W)$ is determined by colouring
strategies that use a finite number of different types of colour classes.
Furthermore, if $W$ is a block graphon with $k\times k$ blocks then $k$ types
of colour classes are sufficient. We also show that if $W$ is block-increasing
or block-Lipschitz then such colouring strategies that use $k$ types determine
the chromatic number up to a multiplicative error of order $O(k^{-1})$.
