From Shopify Wiki

If you buy your domain via Google Apps services (http://www.google.com/apps/), here follows the procedure to link it to your Shopify account. enom (http://www.enom.com) is the hosting provider of Google Apps when you buy a domain through Google.

In Google Apps and enom

1. In your Google Apps interface after login, click on "Domain settings". Some of your account information will show up in a grey square such as your domain name (example.com), your password (16-characters password) and the enom customers services email.

2. In the same grey box, there is a link to the DNS dashboard of the hosting provider (enom). Click on this link (or here: https://access.enom.com/) and login with the your domain name and your password.

3. Once entered the dashboard, scroll down to the Host Records section. A few records must have already appeared such as calendar, docs, mail, sites, start (these are the Google services enabled with your Google Apps account) and all linking to a "ghs.google.com." address.

4. At the bottom of this section, click on the edit button and erase any A-records (usually linking to an IP address).

5. Then click on Add New, and start adding your domain name information.

 If there is an entry under "Host Name" that states: www
 Change the "Address" to your Shopify store name: example.myshopify.com
 Change the "Record Type", and select CNAME(Alias)
 
 If there is no entry with the "Host Name" that states: www
 In "Host Name", put your domain name: www
 In "Address", put your Shopify store name: example.myshopify.com
 In "Record Type", select CNAME (Alias)

6. Repeat the same process, adding a new record for your Root Domain:

 In "Host Name", put '@'
 In "Address", put '204.93.213.45'
 In "Record Type", select 'A record'

In Shopify

Go to your admin screen in Shopify. --> Preferences --> DNS & Domains. Click on “Add a domain you already own” Enter www.example.com; then click on “make this my primary domain”.

And now please wait...

It can often take up to 48 hours for your DNS changes to propagate across the internet, so depending on the situation, you'll need to make sure you've given your domain enough time before you begin advertising it, especially if you've just registered it. In my case, it took less than 10 minutes.

And that's it. You should now be able to visit your store using your shiny new domain name. Now start selling!

If you run into any trouble, you will find it more helpful to first contact Google Apps for support as they have access to those settings (while Shopify doesn't). Here is a direct link to their support page.