To allow an Exchange user to send from an external e-mail address, perform the following steps:

1. Create a Mail Contact with the correct external e-mail address in AD and in Exchange.
2. Open Exchange Management Shell and set up the permissions:

   Get-MailContact MyUser@MyExternalDomain.com | Add-ADPermission -User MyUser@MyLocalDomain -AccessRights ExtendedRight -ExtendedRights "Send As"

   Replace the first e-mail address with the external one, the second part contains the user you wish to assign the permission to.