To delete the currently selected project from the Project Database, click on the button on the Main Menu. You will be asked to confirm your decision.
Note that there is no undo, and once deleted, the project cannot be recovered.
Tip: In an emergency, running an ArcEdit Shell and issuing the oops command may recover a deleted project. However the effects this will have on your Toolbox session are unpredictable, and this procedure is not endorsed by the author.