  Let $X$ be a cubic surface over a global field $k$. We prove that a
Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist
over every extension $L/k$ with degree relatively prime to $3$. In other words,
a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty
Brauer set over some extension $L/k$ with $3\nmid[L:k]$. Therefore, the
conjecture of Colliot-Th\'el\`ene and Sansuc on the sufficiency of the
Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational
point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement
is a special case of a conjecture of Cassels and Swinnerton-Dyer.
