The BIOS should initialize all the video cards it finds so the display should appear on the new card. Most BIOS have settings will allow you to disable the internal peripherals, including the graphics and will also have an option to set which is initialized first. If you add an AGP card, obviously set it to have priority.
There are benefits of not replacing the motherboard:
1. If you use Windows you will have to buy a new license. The new hardware will not match the description MS used to generate your activation key so it will become invalid.
2. Most on-board graphics systems share the memory with the processor, this reduces the amount left for programs and also slows the system down. A plug-in card will have it's own memory and on most modern cards will also have a faster and dedicated graphics processor IC.
3. If it breaks down again you can replace just the graphics card. If the motherboard breaks down you can transfer the graphics card to a new computer.
It's a win-win situation, if you change the motherboard all you do is lose money.
Brian.