First step is to move or remove following type of mailboxes from first database:

1) Any user mailbox
2) Any Archive mailbox
3) Discovery Search mailbox
4) Arbitration mailbox

If you forget to move or remove mailboxes you will get the below error:

The mailbox database "Default DB" cannot be deleted.

Note: I am using Default DB as DB name in this article for First Database. It could be different in your case.

So let’s start moving all of the above mailbox types to another db.

1) Move User Mailboxes:
For these types of mailboxes we can use Exchange Management GUI Console or PS. I prefer EMC as it is easy to use. First login to your Exchange server, launch EMC. Browse to EMC>Recipient Configuration>Mailbox

Create a filter Database>Equal>DBName and Apply filter. This will show you all user mailboxes stored on your first database. Select all mailboxes, right click and click on New Local Move Request and move all mailboxes to another database by following the New Local Move Request wizard.

2) Move Discovery Search Mailbox:
Steps are same as above. It will be named like DiscoverySearchMailbox???

3) Move Archive Mailboxes:
In case you have created any Archive Mailbox on first database. Make sure to move them as well to another database. You can do this with user mailbox move step also.

4) Move Arbitration mailbox:
This one step, where I have seen maximum people struggle to complete and post question on EE site regularly. First these mailboxes are not visible using EMC and can be found using EMS or Exchange management Shell only.

Before, I explain you the steps. First understand these mailboxes are stored at Forest level. So make sure to run this command first.

Set-AdServerSettings -ViewEntireForest $True

Next run below command to view Arbitration mailbox stored on your first database.

Get-Mailbox -Arbitration -Database “Default DB"

Next run below command to move Arbitration mailboxes to another Database.

Get-mailbox –arbitration –database "default db" | new-MoveRequest –targetdatabase "Target DB"

To view the move status run below command

Get-MoveRequest -Sourcedatabase "Default DB"
Or
Get-MoveRequest -Targetdatabase "Target DB"
Or
Open EMC>Recipient Configuration>Move Request

Once all mailboxes are moved and move request shows as completed 100%, you can clear the all move requests and then you can dismount/remove the database.

For dismounting and removing DB. Goto EMC>Organization Configuration>Mailbox>Database Management

Select the first database>Right Click Dismount then Right click again and Select Remove.

Finally, browse to db path in your machine and delete the edb and logs files.

I hope this article will help everyone. Please let me know your comments or suggestions.