Consider the reduced free product of C*-algebras, (A, ) = (A1, 1) * (A2, 2), with respect to states 1 and 2 that are faithful. If 1 and 2 are traces, if the so-called Avitzour conditions are satisfied, (i.e. A1 and A2 are not "too small" in a specific sense) and if A1 and A2 are nuclear, then it is shown that the positive cone, K0(A)+, of the K0-group of A consists of those elements g K0(A) for which g = 0 or K0()(g) > 0. Thus, the ordered group K0(A) is weakly unperforated. If, on the other hand, 1 or 2 is not a trace and if a certain condition weaker than the Avitzour conditions holds, then A is properly infinite.
