Setting up the new domain (newcustomdomain.com)
1. Go to Blogger Dashboard, 'Settings', 'Basic'. Under Publishing, 'Blog Address', click on 'edit' next to oldcustomdomain.com address.
2. If you already own newcustomdomain.com, under 'Advanced Settings' type newcustomdomain.com, click the 'Redirect newcustomdomain.com to www.newcustomdomain.com' and hit Save.
3. If you don't own it yet, click on 'Need a domain? Buy one now' then do Step 2.
Redirecting the old domain (oldcustomdomain.com) to the new domain (new customdomain.com)
1. Assuming you bought your old domain though blogspot, you will need to login to ENOM (access.enom.com) to change your 'Host name' details. If you don't know your username or password, sign in to manage oldcustomdomain.com through Google Apps (http://www.google.com/enterprise/apps/business/). On your Dashboard, click on 'Domain Settings', 'Domain names', 'Advanced DNS Settings' and make a note of the domain and password.
2. Login with you old domain and password on access.enom.com
3. Click on 'Host records', 'edit'.
4. Erase everything written on both 'Host Name' and 'Address' fields.
5. On the first 'Host name' line type 'www'. On the first 'Address' line type the full URL of your new domain (http://www.newcustomdomain.com/). On the first 'Record Type', select 'URL redirect'.
6. On the second 'Host name' line type '@'. On the second 'Address' line type the full URL of your new domain (http://www.newcustomdomain.com/). On the second 'Record Type', select 'URL redirect'.
7. Hit save.
I recommend you check that the links are being redirected properly before you close the ENOM website. It can take a few minutes for it to update and you may see an error message when you type your old domain (oldcustomdomain.com), but after everything is up and running, whenever you type your old domain, it will take you straight to your new domain.
I can't guarantee it will work for you, but it worked perfectly for me. I had to search through a lot of info online to find a way to make it work for me, so all I listed there is based on my own experience.
If you have questions feel free to drop me an e-mail at firstname.lastname@example.org.
I hope this helps!
Lots of love,