Let $\chi$ be a totally odd character of a totally real number field. In 1981, B. Gross formulated a p-adic analogue of a conjecture of Stark which expresses the leading term at s=0 of the p-adic L-function attached to $\chi\omega$ as a product of a regulator and an algebraic number. Recently, Dasgupta-Darmon-Pollack proved Gross' conjecture in the rank one case under two assumptions: that Leopoldt's conjecture holds for F and p, and a certain technical condition when there is a unique prime above p in F. After giving some background and outlining their proof, I will explain how to remove both conditions, thus giving an unconditional proof of the conjecture. If there is extra time I will explain an application to the Iwasawa Main Conjecture which comes out of the proof, and possibly make some remarks on the higher rank case.