Even with a 1GB network in place you've going to run for a very long time.
Our Data Centre runs ARCserve (couldn't put in Backup Exec due to Itaniums running!). Our SAP DB is huge and runs across a SAN, which takes 12 hours.
You'd have to look at trying to make your backups run a lot simpler:
1. Small SAN if you can afford it. You'd be taking advantage of 4GB/s throughput meaning your backups would actually finish before your next backup window starts.
2. Checking if you're running 100MB/1GB. See if you can change to 1GB and then start your backup as soon as it's possible according to business requirements. Change your NIC speeds to 1000MB FULL, and do the same on the switches to get the best speed out of them.
3. See if you can change the way in which your backups run. Try Differential backups, with your FULL backup running over the weekend.
4. Seeing if Backup Exec can do a multiplexing job (send multiple streams to the drives).
5. Change the location of the library physically to the server in question if it's not there already. Running a backup across a LAN is far slower than a local backup.
How big is your daily job? 7TB as well? If it's all going through SCSI, and the same SCSI card at that, it's going to take as long, if not longer, no matter if you backup to different drives.