There are two free website builders allowing to connect a custom domain name at no charge.
ConvertKit is actually a newsletter tool that allows you to create free landing pages, which is a one-page website. If you already have a custom domain, they let you connect it for free.
Shift4Shop is an ecommerce store builder. They let you build an online store for free and also connect a domain name. The way they make money is through processing fees and upgrades to premium plans.