Answer:
Presumably the LLC has a provision in its membership agreement relating to what happens if the single member dies. Assuming that the LLC can legally continue in existence with a new single member, it would have to either elect not to continue being disregarded by the IRS as an entity (and apply for an EIN for the LLC), or start using the TIN of its new single member.
It cannot continue to use the deceased individual's SSN. That number can only be used after his/her death in connection with filing a final tax return.
First published on BankersOnline.com 3/15/10