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Matches is right. Concrete is a pourous material so it pretty-much can absorbe a good bit of any liquid that's left to 'sit' on it over time.
You could mix a solution of 4 to 1 (water to bleach) and 'paint' it on. Just be sure to keep it wet so the bleach gets a chance to work itself into the concrete. Cover the area with a good-sized sheet of plastic while it's doing its thing. This way, it won't evaporate so quickly. Then, at some point, you need to give up and just go ahead and lay your new floor. You could consider sealing that area after it dries. Visit your local Paint Retailer (found in the YELLOW PAGES under PAINT - RETAIL) for a sealer. Just tell them what you're interested in doing so they give you the 'right stuff'.
My best to ya and hope this helps.
Jay J -Moderator
PS: God Bless America!