  We combine density functional theory and scanning tunneling microscopy to
study the proximity effects between a bulk Ru(0001) superconductor and an
atomically thin overlayer of Co. We have identified that the Co monolayer can
grow in two different stackings: the hcp and a reconstructed \textepsilon-like
stacking. We analyze their electronic structure from both experiments and
density functional theory. While the magnetic hcp stacking shows a weak
proximity effect in combination with Shiba states and with almost no
suppression of superconductivity of the substrate, the more complex
\textepsilon-like stacking becomes almost fully superconducting and displays an
edge state at the island rim. We identify this edge state as a trivial state
caused by a local hcp rim around the \textepsilon-core. We explain the weak
proximity effect between Ru and the magnetic hcp islands by a low transparency
of the interface, while the large chemical unit cell of the non-magnetic
\textepsilon-like stacking lifts the momentum conservation at the interface
making it transparent and causing a clear proximity effect.
