It seems to be an exception in that other words with the same spelling don't have this vowel sound. cut /kʌt/ nut /nʌt/ gut /gʌt/ hut /hʌt/ jut /dʒʌt/ rut /rʌt/ all have the same phoneme, while only 'put' in different: Put is pronounced put /pʊt/ to rhyme with foot /fʊt/. It's the same sound as in book /bʊk/, hood /hʊd/ and so on, so logically 'put' should be spelt 'poot'. Unfortunately it isn't, so yes, it's an exception to the rule.

As far as I have been speaking English myself, I have always pronounced it "poot". No exception! Beware that the "oo" sound is quite short anyway :)