From Prof. Volokh, we learn that Congress has passed its "Saxbe fix." This should protect Sen. Clinton from a successful challenge to her Secretary of State nomination based on the emoluments clause* of the United States Constitution.
Note the word "successful" in the previous sentence. Someone's going to file a lawsuit about this. And the lawsuit will find its way to the Supreme Court as a cert. petition. And there might be four votes to hear it. It seems, however, too bizarre to imagine that there would be five votes for nullification of Senator Clinton's appointment.**
*"No Senator or Representative shall, during the time for which he was elected, be appointed to any civil office under the authority of the United States, which shall have been created, or the emoluments whereof shall have been increased during such time: and no person holding any office under the United States, shall be a member of either House during his continuance in office."
**Those counting votes at home should consider this wikipedia article, which implies that Justice Kennedy owes his seat on the Court to the emoluments clause. See Michael Scott's brilliant thoughts on Wikipedic reliability here.