When the last all girls Catholic School in our diocese closed, a group of parents got together and with approval, formed a new school but as a seperate entity. This school is in the within the diocese but not a diocessen school. I believe it has its own board of directors.
To control costs, they started out by renting space in a church parish center, but as their enrollment has grown they have moved into a former parish school building.
If you want more info from them, send me a PM.