Exchange cannot send messages with FullAccess and SendAs permissions

MS Exchange 2010 SP1, version 14.3 (Build 123.4). A user has a mailbox and an AD account. The user needs a second address with a separate mailbox. We created a separate mailbox with a separate disabled AD user and granted the user Full Access permissions to it. The Hide from Exchange address lists option was enabled for the second mailbox. Sending messages as that mailbox then failed:

Delivery to these recipients or groups could not be completed:

[Recipient name]
You cannot send a message on behalf of another sender without permission to do so.
Make sure the message is sent on behalf of the correct sender or request the required permission.
If the problem persists, contact support.

Solution: do not enable the Hide from Exchange address lists option.

Second part of the solution: the address used by the Exchange user to send mail must be present in the Offline Address Book. In this case, the OAB had not yet updated and SendAs returned an error.