The sector of analyticity of the Ornstein–Uhlenbeck semigroup is computed on the space $L^p_\mu \,{:=}\, L^p (\R^N; \mu)$ with respect to its invariant measure $\mu$. If $A\,{=}\,\Delta + Bx\cdot \nabla$ denotes the generator of the Ornstein–Uhlenbeck semigroup, then the angle $\theta_2$ of the sector of analyticity in $L^2_\mu$ is ${\pi}/{2}$ minus the spectral angle of $BQ_\infty, Q_\infty$ being the matrix determining the Gaussian measure $\mu$. The angle of analyticity in $L^p_\mu$ is then given by the formula \[\cot {\theta_p} = \frac{\sqrt{(p-2)^2+p^2(\cot\theta_2)^2}}{2\sqrt{p-1}}.\]