  The chromatic number $\chi(G)$ of a graph $G$ is defined as the minimum
number of colours required for a vertex colouring where no two adjacent
vertices are coloured the same. The chromatic number of the dense random graph
$G \sim G(n,p)$ where $p \in (0,1)$ is constant has been intensively studied
since the 1970s, and a landmark result by Bollob\'as in 1987 first established
the asymptotic value of $\chi(G)$. Despite several improvements of this result,
the exact value of $\chi(G)$ remains open. In this paper, new upper and lower
bounds for $\chi(G)$ are established. These bounds are the first ones that
match each other up to a term of size $o(1)$ in the denominator: they narrow
down the colouring rate $n/\chi(G)$ of $G \sim G(n,p)$ to an explicit interval
of length $o(1)$, answering a question of Kang and McDiarmid.
