Before you do it
You will have to make sure that both Apache web servers are the same version, I didn’t and it caused some problems.
My hosting company also mentioned that both cPanels must be the same version too, but I’m not sure about this.
You will also need root access to both servers and an SSH client like putty, so go ahead and connect to both servers.
Manually Moving WHM/cPanel accounts
Follow these steps:
1 – In your old server do:
/scripts/pkgacct “username” <------- Don't type the quotes
2 - After the first step is done do:
scp /home/username.tar.gz root@IP:/home/ <------- Where IP is the IP of your destination server
You will be asked for the root password.
3 - In your destination server do:
cd /home
/scripts/restorepkg “username” <------- Don't type the quotes
Disclaimer
Do this at your own risk, it worked fine on my version of WHM/cPanel/CentOS.

I'm a programmer at 
That was helpful.
You can do it from backup/restore function as well! It is much easier for non techie people.
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backup/restore times-out on very large accounts and the process ends up being incomplete…
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Yup,
I do agree with you.
In that case, it’s really helpful.
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How would you transfer whm/ cpanel from one domain name to the another?
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Make a new account in the destination server and use WHM’s account transfer and choose the new account as the target…
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