DO NOT DO NOT give up the old card until you have the new one. You don't have to give it up no matter what they say.
RM 00201.080 Field Office (FO) Disposal of Returned Social Security Number (SSN) Cards:
http://web.archive.org/web/20060210014436/.../lnx/01002010802. Card Returned With Request For Replacement Card or New SSN
If the applicant has a card and requests a replacement card (duplicate or corrected) or a new SSN:
-- Ask the applicant to return the old SSN card. (If the applicant does not have the card with him or her, remind the applicant to return the old card to SSA for destruction or to keep the old card in a secure place [not in a wallet or purse]).
NOTE: Do not forcefully confiscate the SSN card if the applicant refuses to relinquish it. Similarly, return the old SSN card if the applicant requests that it be given back.