In this article, we study the transcendental part t_2(S) of the Chow motive of a smooth projective surface S over a field. It is defined as a certain direct summand of h_2(S). The main results are two different formulas for the endomorphism ring of t_2(S). As a corollary, we find that this endomorphism ring remains the same when one passes to the category of "birational Chow motives" (see #596 of this archive), and even to a smaller category defined with "correspondences at the generic point". Conjectures of Beilinson and Murre imply that End(t_2(S)) should also be isomorphic to the endomorphism ring computed in the category of motives modulo homological equivalence (for a classical Weil cohomology theory), and in particular should be finite dimensional over Q. Relationships with S.I. Kimura's notion of finite dimensionality and with Voevodsky's triangulated categories of motives are also given. Although no progress on Bloch's conjecture on surfaces is made in the paper, that conjecture was the starting motivation for writing it. It is to appear in the proceedings of the conference in honor of J.P. Murre's 75th birthday.