To encourage this, Blogger recently added the "Buy A Domain For Your Blog" wizard. Like all Blogger wizards, it has its advantages and its disadvantages.
Note: In the examples below, I'm using the "ping" command to diagnose my example domain, mydomain.com. You may check your specific domain, similarly.
To use the wizard, simply go to the dashboard, and to Settings - Basic, and click on "Add a custom domain". Here, you can purchase the domain (when available) of your choice, and simply tell Blogger (and the "partner" registrar) that you want your blog published to "www.mydomain.com" ("mydomain.com" being the domain of your choice).
The wizard completed, and your credit card debited, you'll have your blog now responding to "www.mydomain.com", with "myblog.blogspot.com" redirected to "www.mydomain.com". And "mydomain.com" will go to a Google Apps server (and then to "www.mydomain.com").
C:\>ping mydomain.com Pinging mydomain.com [126.96.36.199] with 32 bytes of data: Reply from 188.8.131.52: bytes=32 time=243ms TTL=245 Reply from 184.108.40.206: bytes=32 time=229ms TTL=245 Reply from 220.127.116.11: bytes=32 time=228ms TTL=245 Reply from 18.104.22.168: bytes=32 time=232ms TTL=245 Ping statistics for 22.214.171.124: Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds: Minimum = 228ms, Maximum = 243ms, Average = 233ms C:\>ping www.mydomain.com Pinging ghs.l.google.com [126.96.36.199] with 32 bytes of data: Reply from 188.8.131.52: bytes=32 time=214ms TTL=244 Reply from 184.108.40.206: bytes=32 time=199ms TTL=244 Reply from 220.127.116.11: bytes=32 time=225ms TTL=244 Reply from 18.104.22.168: bytes=32 time=219ms TTL=244 Ping statistics for 22.214.171.124: Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds: Minimum = 199ms, Maximum = 225ms, Average = 214ms
This reflects a pair of DNS records for mydomain. The primary domain directed to a trio of Google Apps servers.
mydomain.com. 3600 IN A 126.96.36.199and the "www" alias directed to Google
mydomain.com. 3600 IN A 188.8.131.52
mydomain.com. 3600 IN A 184.108.40.206
www.mydomain.com. 3600 IN CNAME ghs.google.comand your blog "myblog.blogspot.com" is now published to "www.mydomain.com". All of this in one transaction, and everything directed properly. Nice.
But note that you will experience a 72 hour period, after setting the domain up, when there will be some vagaries.
For a more detailed description of the above process, see my chronicle Using The "Buy A Domain For Your Blog" Wizard.
If you get the well known "Another blog ..." error, you can deal with that using Google Apps.