I will qualify my earlier response a bit. From the interagency FAQ on CIP requirements
, question 2 under "Information required," reads:
2. Can a bank open an account for a U.S. person that does not have a taxpayer identification number? No, the bank cannot unless the customer has applied for a taxpayer identification number, the bank confirms that the application was filed before the customer opened the account, and the bank obtains the taxpayer identification number within a reasonable period of time after the account is opened. Note, however, that a bank does not need to obtain a taxpayer identification number when opening a new account for a customer that has an existing account, as long as the bank has a reasonable belief that it knows the true identity of the customer. A bank may also open an account for a person who lacks legal capacity with the identifying information, including taxpayer identification number, of an individual who opens an account for that person.
The bold text is an exception available only if the person has an existing account with you, and the bank reasonably believes it knows his true identity.
Without an SSN, however, you will be required to do backup withholding on any interest paid on the new account.