Make 2 programs. Your primary application, and a launcher. The launcher is what people should click on to start your program, and it is responsible for starting the primary application, and then can shut itself down. If you need to update while the primary application is running, you can download the updated program to a separate file, along with any other files necessary. Then the primary application can launch the launcher and shut itself down. The launcher then is responsible for deleting the old program and renaming the new one, and relaunching it.
Actually, this doesn't even have to be two different programs. It can be the same program operating in two different modes (selected through command line arguments, for example). You still need 2 copies of the executable file on disk though. Note that this is precisely what Google Chrome does.