There are two factors.
In practical terms, who possesses the files and access credentials necessary to maintain the site? It sounds like it isn't you, so in turn it depends on who registered the domain and who controls the access with the web site host.
Secondly, the web site designer owns the copyright unless you have a written transfer of ownership of that copyright.
Armed with this knowledge you might want to make a new decision about creating a site and having it hosted.