  We prove that if $L$ is a non-trivial alternating link embedded (without
crossings) in a closed surface $F\subset S^3$, then $F$ has a compressing disk
whose boundary intersects $L$ in no more than two points. Moreover, whenever
the surface is incompressible and $\partial$-incompressible in the link
exterior, it can be isotoped to have a standard tube at some crossing of any
reduced alternating diagram.
