No. The solicitor must not divulge to anyone the details that a client has given him. That information is between him and his client and subject to legal professional privilege - only the client can waive that privilege. If this matter went to trial, the solicitor could continue to act but could not put forward a positive defence (thus the defence would consist of testing the prosecution's case, rather than putting forward say, an alibi). In fairness however, most solicitors would decline to act on the basis that they are professionally embarrassed. In my experience, at the slightest hint of any such embarrassment most lawyers withdraw immediately, and properly so.