If your A record points to your droplet, typing in the domain in your browser should lead you to DO. But if you've just changed your DNS records, it may take a while before the change takes effect: the update has to spread to ISPs throughout the world. It can take up to 24 hours. You can test where the domain name is currently pointing by pinging the domain and seeing which IP address responds.

I guess I didn't ask my question carefully enough. Sorry. I'll try again.

My sites now all point to my content on Digital Ocean. It's working.

But, the DNS itself is hosted with 1and1.com. I am moving away from 1and1.com and I don't want them to host the DNS anymore. I'd rather have Digital Ocean host the DNS.

Domain Name Transfer is a well-known and common activity. I want to transfer the hosting of my Domains to Digital Ocean, not just point the domains hosted at 1and1.com to my Digital Ocean IP address, which I have already successfully done.

Can this be done? How? It isn't in the FAQ.

You can't transfer your domain to DO. You can either

1. Use 1and1 to point to your droplet. Like you already have.
2. Transfer you nameservers to DO. Then create A CNAME MX records on DO DNS control panel. See https://www.digitalocean.com/community/tutorials/how-to-point-to-digitalocean-nameservers-from-common-domain-registrars You will still renew and maintain you domain at 1and1 as the registar. Don't think this is what you want.

Reading the above it looks like you want to part ways completely with 1and1. You will need to transfer the domain to a different registar. Personally I like namecheap, you just missed the move your domain day deal for $3.98

Thank you kindly. You are exactly correct that I would like to sever my ties with 1and1.com. I'll spare you the painful details, but I moved all my sites to DO today on the recommendation of a friend. So far so good and I'm very happy with the value. I'll consider namcheap -- too bad I missed the deal, but good to know they have deals like that sometimes. :-)