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Dominating Set is the problem of deciding, given a graph $G$ and an integer $k\\geq 1$, if $G$ has a dominating set of size at most $k$. It is well known that this problem is $\\mathsf{NP}$-complete even for claw-free graphs. We give a complexity dichotomy for Dominating Set for the class of claw-free graphs with diameter $d$. We show that the problem is $\\mathsf{NP}$-complete for every fixed $d\\ge 3$ and polynomial time solvable for $d\\le 2$. 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Dominating Set is the problem of deciding, given a graph $G$ and an integer $k\\geq 1$, if $G$ has a dominating set of size at most $k$. It is well known that this problem is $\\mathsf{NP}$-complete even for claw-free graphs. We give a complexity dichotomy for Dominating Set for the class of claw-free graphs with diameter $d$. We show that the problem is $\\mathsf{NP}$-complete for every fixed $d\\ge 3$ and polynomial time solvable for $d\\le 2$. 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