Hi Paul -
Unfortunately, the words "Breakers" and "cheap" do not go together: it is THE legendary luxury hotel in West Palm Beach, one of the best in the world, and is going to be VERY expensive any way you book it. Just plugging in a random January weeknight, Kayak.com and Bookingbuddy.com turn up rates running between550 and620 a night on Orbitz, Travelocity. Expedia, Hotwire, Hotels.com, Venere (not sure these rates include taxes).
So the best thing you can do is to plug your dates into your preferred hotel search sites and see what turns up. And, of course, you can always contact the hotel but in this case, I think you might find the best rates on one of these comparative sites.
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