Use the WHM account transfer function. If you don't have root to your old machine, use "Copy an account from another server with account password" in the Transfers section of WHM. All you need to do is put the IP address from the old server, the domain name, user name and account password and WHM will import everything, create the account and all. Then you just need to update your DNS (if you changed DNS).
I've transferred about 60 accounts (even one that was 948 MB) using this method.