If the lawyer knows in advance that the communications are privileged, he is not to use them. Consider a DC Bar ethics opinion on a similar situation:
Where a lawyer has inadvertently included documents containing client secrets or confidences in material delivered to an adversary lawyer, and the receiving lawyer in good faith reviews the documents before the inadvertence of the disclosure is brought to that lawyer’s attention, the receiving lawyer engages in no ethical violation by retaining and using those documents. Where, on the other hand, the receiving lawyer knows of the inadvertence of the disclosure before the documents are examined, Rule 1.15(a) requires the receiving lawyer to return the documents to the sending lawyer; the receiving lawyer also violates Rule 8.4(c) if the lawyer reads and/or uses the material.
Ethics opinion 256. In a similar vein is NY City Bar Formal Opinion 2003-04:
QUESTIONS: What are the ethical obligations of the lawyer who receives a misdirected communication containing confidences or secrets? Must the lawyer notify the sender? Must the lawyer return the communication and/or destroy all copies? May the lawyer review the communication? May the lawyer use information learned from the communication?
OPINION: A lawyer who receives a misdirected communication containing confidences or secrets should promptly notify the sender and refrain from further reading or listening to the communication, and should follow the sender's directions regarding destruction or return of the communication. However, if there is a legal dispute before a tribunal and the receiving attorney believes in good faith that the communication appropriately may be retained and used, the receiving attorney may submit the communication for in camera consideration by the tribunal as to its disposition. Additionally, the receiving attorney is not prohibited as an ethical matter from using the information to which the attorney was exposed before knowing or having reason to know the communication was inadvertently sent. However, it is essential - as an ethical matter - that the receiving attorney promptly notify the sending attorney of the disclosure in order to give the sending attorney a reasonable opportunity to promptly take whatever steps he or she feels are necessary.
While they deal with inadvertent disclosure directly to the attorney, I see no logical reason that the attorney’s ethical obligations should be any different when the confidential communication is provided to him/her indirectly by his/her client under circumstances in which the client should not have obtained that communication (e.g. snooping into mail not addressed to her).