If H is an abelian extension of an imaginary quadratic field K, then the group of units of H contains an important subgroup consisting of modular units. These are obtained as special values of certain modular functions, and are analogous to the cyclotomic units of an abelian extension of Q. Suppose that H is an abelian extension not only of K but of Q as well. In this case, H contains both cyclotomic units and modular units, and the question arises as to the relation of these two groups of units. The study of this situation was begun by Kubert-Lang [KL 1] in the case K = Q(1/-p), H = Q(p,) (if p- 3 mod 4), or Q(p,,) (if p _ 1 mod 4). In this paper, we consider the general case of an arbitrary abelian extension of Q containing an imaginary quadratic field. We show that, for some power 21 of 2, the 21th power of the modular unit group of H is contained in the cyclotomic unit group of H. The exponent X depends only on the imaginary quadratic field K. We do this by deriving explicit formulae expressing modular units as power products of cyclotomic units. In Section 1 we define the two unit groups with which we will be dealing. We also describe more explicitly the fields H which are simultaneously abelian over Q and K. In Section 2, we use a factorization of L-series to obtain a set of linear relations between the logarithms of the modular units of H and those of the cyclotomic units of H. In the next two sections, we show how to solve these relations to express the modular units in terms of the cyclotomic units. At this point, we have to consider two separate cases. We treat the first, which is much simpler, in Section 3, providing a pattern for the more involved case of Section 4. In Section 5 we show that the exponents occurring in the products derived in Sections 3 and 4 are actually integral. Following a suggestion of the referee, we have replaced our original proof, which was by direct calculation, and have instead deduced this from the fundamental Theorem