Apparently you can - Azure now supports A Records as it provides fixed IP addresses:
With an A record, you map a domain (e.g. contoso.com or
www.contoso.com) or a wildcard domain (e.g. *.contoso.com) to the IP
address of a deployment within a Windows Azure hosted service.
Accordingly, the lifetime of this IP address is the lifetime of a
deployment within your hosted service.
The IP address gets created the first time you deploy to an empty
slot (either production or staging) in the hosted service and is
retained by the slot until you delete the deployment from that slot.
See this link for more information - Section: Expose Your Application on a Custom Domain
Just tested it with one of my own domains and a sample Azure app in staging. - Works a treat!
I found when you use the Swap VIP option, to move from staging to production, your staging IP address gets taken with you. This meant I retained the staging website pointed at by my A record after republishing and after swapping to Production. The downside to this is that your domain points at staging or at production when you swap, which is probably not what you wanted as your domain would likely only point at production. A CNAME record will do that, but does not allow for a blank subdomain.