  The diamond is the graph obtained by removing an edge from the complete graph
on 4 vertices. A graph is ($P_6$, diamond)-free if it contains no induced
subgraph isomorphic to a six-vertex path or a diamond. In this paper we show
that the chromatic number of a ($P_6$, diamond)-free graph $G$ is no larger
than the maximum of 6 and the clique number of $G$. We do this by reducing the
problem to imperfect ($P_6$, diamond)-free graphs via the Strong Perfect Graph
Theorem, dividing the imperfect graphs into several cases, and giving a proper
colouring for each case. We also show that there is exactly one
6-vertex-critical ($P_6$, diamond, $K_6$)-free graph. Together with the
Lov\'asz theta function, this gives a polynomial time algorithm to compute the
chromatic number of ($P_6$, diamond)-free graphs.
