a {\em noncommutative projective surface} is a noetherian graded domain of gelfand-kirillov dimension 3; their classification is one of the most important areas of research in noncommutative algebraic geometry. we complete an important special case by classifying all noncommutative projective surfaces that are {\em birationally commutative}: to wit, they are graded subrings of a skew polynomial ring over a field. we show that birationally commutative projective surfaces fall into four families, parameterized by geometric data, and we obtain precise information on the possible forms of this data. this extends results of rogalski and stafford on rings generated in degree 1, although our proof techniques are significantly different.