Hi Gaston,
Are you using Office 365 for Professionals and Small Businesses? From your post, I understand that you want to move your email to Office 365 and keep your current public-facing website. Have I misunderstood anything?
Since the problem may caused by incorrect DNS records, please refer to the following information to check and change your DNS records.
Firstly, I’d like to confirm that if you have changed the name servers to Office 365. The link below can provide you steps about how to change the name servers to Office 365 and add an A record to keep your website with your current hosting provider:
onlinehelp.microsoft.com/.../hh335299.aspx
Alternatively, if you don’t want to change name servers of your custom domain to Office 365, you can also use your domain name with Office 365 without redirecting your name server records to Office 365. However, you need to configure the required DNS records yourself, and there will be some limitations in Office 365 services when Office 365 doesn’t host your DNS.
For more information, please refer to the following article:
onlinehelp.microsoft.com/.../hh416759.aspx
Thanks,
Nancy Xie