The Borodin--Kostochka conjecture states that every graph $G$ with maximum degree $Δ(G)\ge 9$ satisfies $χ(G)\le \max\{ω(G),Δ(G)-1\}$. In this paper, we verify this conjecture for graphs with sufficiently large maximum degree. More precisely, we prove that every graph $G$ with maximum degree $Δ\ge 5.3\times 10^6$ and clique number $ω(G)<Δ$ satisfies $χ(G)\le Δ-1$. This improves a longstanding result of Reed.