Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by a nonmaximal order. Gross' Conjecture has been reformulated in the language of derived categories and determinants of perfect complexes. Burns and Flach have realized that this immediately leads to a refinement of Gross' Conjecture. The conjecture is now expressed as a statement concerning a generator of the image of a map of 1-dimensional modules. This conjecture is proved by a construction which shows it to follow from the Explicit Reciprocity Law and Rubin's Main Conjecture.