Yeah I'm afraid you're going to need to get either a new computer or a new graphics card - alas both aren't exactly cheap options. You could have a look for a pre-flashed 6800 card. There's a flashed XFX 6800 on eBay at the moment if you're in the UK. Chances are it won't go for masses. Genuine Apple 6800s go for way more.
If you're already looking at spending a good chunk on a new graphics card you may want to consider selling the Mac Mini and buying one of the new gen Mac Minis and a mini display port to dual link adapter, that will be able to drive the 30" just fine.
I have to ask though, why would you want to use a 30" ACD as a 'TV'? If you already had it connected to the G5 and wanted to carry on using it then I'd get it but it sounds like you've got a 'spare' 30" ACD which you don't need elsewhere anymore. If that's the case then I'd highly suggest you sell it pronto (they still command a hefty premium second hand). For example, looking at completed listings in the UK, you could get £600 at the very least, probably more. That money can easily buy you a great 40" Full HD LCD TV brand new. If you're elsewhere in the world then I'm sure you could get similar deals.
Don't get me wrong, the 30" ACD is an absolute dream and is in my opinion the best consumer screen available on the market - as far as computer screens go but it's not as good a TV as an actual TV, it's built for different purposes after all. If you want to go the media centre route then a proper TV + your Mac Mini running Plex would be absolutely perfect. It's what I've got in my lounge.