If it's coming off in full pieces, most likely the substrate is at fault
so the adhesive doesn't have a good surface to bond to.
As has been said, contact cement is the proper adhesive but to get good
adhesion over a previous installation will have to clean and prep both
surfaces to be rid of old cement and be dirt and particularly
grease-free. If the counter edge is not smooth (like a flaky particle
board, perhaps), fill it w/ a hard filler and resurface (a flat wood
file is good for this rather than sanding as it will keep the surface
flat rather than rounding it over as sanding tends to do) it.
Follow instructions about application on the container of contact cement
carefully -- in particular, on the counter surface it will undoubtedly
take at least a second coating as the first will be absorbed into the
material (again assuming it's particle board or similar).
My preference is still to use the original solvent-based contact cements
despite the odor and flammability issue--to me they seem to work better
than the water-based.
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