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We study the Ornstein-Uhlenbeck semigroup $P(t)$ associated with the Ornstein-Uhlenbeck operator $$ Lf(x) = \\frac12 {\\rm Tr} Q D^2 f(x) + <Ax, Df(x)>.$$ Here $Q$ is a positive symmetric operator from $E^*$ to $E$ and $A$ is the generator of a $C_0$-semigroup $S(t)$ on $E$. Under the assumption that $P$ admits an invariant measure $\\mu$ we prove that if $S$ is eventually compact and the spectrum of its generator is nonempty, then $$\\n P(t)-P(s)\\n_{L^1(E,\\mu)} = 2$$ for all $t,s\\ge 0$ with $t\\not=s$. This result is new even when $E = \\R^n$. 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