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In your public DNS, add a CNAME record to point to the azure DNS record. For example, in the above example, I have a DNS in Azure for the VM of jiwards.australiaeast.cloudapp.azure.com. I want my users to use jiwards.jiwa.com.au to reach the machine, so I enter a DNS CNAME record in my DNS provider (Cloudflare in this example) which maps jiwards.jiwa.com.au to jiwards.australiaeast.cloudapp.azure.com, and without the need for the VM to have a static IP address:
We won't be able to connect using that DNS name just yet - the *.australiaeast.cloudapp.azure.com domain will need to be used until we create the necessary SSL certificates.
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