Let \(G\) be a claw-free graph with independence number \(\alpha(G) \geq 3\). Then, it is shown that the chromatic number \(\chi(G) \leq 2\omega(G)\), the clique number. This is an analogue of Vizing's Theorem for line graphs \(H\) of simple graphs that \(\chi(H) \leq \omega(H) + 1\). Examples are given to show that the bound \(2\omega(G)\) is sharp for \(\chi(G)\), but if the condition \(\alpha(G) \geq 3\) is dropped, the bound for \(\chi(G)\) is not linear in \(\omega(G)\). The choice number \(ch(G)\), which is a list coloring number, is also considered, and it is shown that \(ch(G) \leq 2\omega(G)\) for claw-free graphs with \(\alpha(G) \geq 3\), except for possibly a special class of claw-free graphs (Schläfli graphs, which are described in a series of articles of the authors characterizing claw-free graphs [``The structure of claw-free graphs,'' Surveys in combinatorics 2005. Papers from the 20th British combinatorial conference, University of Durham, Durham, UK, July 10--15, 2005. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 327, 153--171 (2005; Zbl 1109.05092); ``Claw-free graphs. I: Orientable prismatic graphs,'' J. Comb. Theory, Ser. B 97, No.\,6, 867--903 (2007; Zbl 1128.05031); ``Claw-free graphs. II: Non-orientable prismatic graphs,'' J. Comb. Theory, Ser. B 98, No.\,2, 249--290 (2008; Zbl 1137.05040); ``Claw-free graphs. III: Circular interval graphs,'' J. Comb. Theory, Ser. B 98, No.\,4, 812--834 (2008; Zbl 1158.05035); ``Claw-free graphs. IV: Decomposition theorem,'' J. Comb. Theory, Ser. B 98, No.\,5, 839--938 (2008; Zbl 1152.05038); ``Claw-free graphs. V. Global structure,'' J. Comb. Theory, Ser. B 98, No.\,6, 1373--1410 (2008; Zbl 1196.05043)]).