Yes. Your hosting works with a domain you already own, and the domain stays where it is, with the company you bought it from.
When you order, you enter that domain name. Nothing happens to the domain at that point. It keeps working as it does today until you change its settings to point at your hosting account.
- Your registrar, the company you bought the domain from, holds the domain name. You renew it with them. It is not part of your hosting invoice.
- Up At Last hosts the website and, if you want, the email for that domain.
Keep your registrar login somewhere safe and keep the domain's renewal up to date. If the domain lapses, the site and email stop working no matter how healthy the hosting is.
If you do not have a domain yet
Up At Last does not sell domain names, so you need to get one from a domain registrar before you order. Check the renewal price, not only the first-year price, before you buy.