PICO is correct. I am having the same problem going from a drop in to an under mount. Unfortunately my laminate counters will have to be replaced but we don't have this expense in our budget. We are thinking of trying to paint and seal the raw cut out edge with some kind of marine spar varnish. This might be something you could try. Be sure to use the marine varnish like they use to seal boat decks and do several coats. We found the sink for a bargain at a yard sale. It's an expensive stainless one that retailed for well over $300 and we got it for $25. It wont be such a deal if we have to spend a lot replacing the counter tops, tho!
You will have to install a new counter top too. The edges of the counter top where the sink is set in will not be finished. The edges will be wood so when you put the sink underneath your counter you will see the wooden edge. I don't know about marble or granite tops, but laminate or ceramic will have a wood base.
When your old countertop was originally created, a hole was cut to put the sink into, and it is likely that the vertical surfaces would not have been finished. They're likely to be rough (plywood showing).
You will have to see what kind of condition your countertop is in underneath the old sink. You should be able to mount it under your old counter. You could go to your local hardware store and ask someone there or go online under some do it yourself websites.
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