When a member joins and their account is created, a photo album for their profile is created. If you are not planning on using profile photos, meaning you won't have the profile photo block on the profile page, then you could use that album for the single album. You would need to hack the code and remove the bits that allow them to create a new album. The upload button is just one place for creating a new album. I have no idea how involved this would be and what problems this could introduce.
The other solution would be to code in a limit on the number of photo albums that one can create and this would be the more elegant solution. You could leave the profile photos in place for you members to use on their profile page. This solution may not be that hard to code in, since Dolphin already tracks the number of albums a member has.; I am sure you have seen the message, "You have 3 albums, click here to create a new one"; instead, it could say, "You have 3 albums and have reached your limit." It would be nice if this would have a setting in the backend for admins but hardwiring it in is less work. I am not sure of all the entry points to creating a new album; one is the pop-up and pop-ups are handled by different code.
The bottom line is that what you want to accomplish is not already a part of Dolphin.