As Pokemon Red/Blue/Yellow/Green/Gold/Silver/Crystal were designed for the Game Boy & Game Boy Color, they were not designed for the GBA Hardware. As such, they cannot detect GBA Hardware, including the link-cables. You will require the Game Boy Color generation cable when trading over using any compatable GBA format.
The reason the GBA plays Game Boy/GBC games, is because there's a physical switch inside the cart slot that tells the GBA to switch to GBC mode, when a GB/GBC cartridge is inserted.
GBA systems compatable with GB / GBC games are;
GBA classic GBA SP Game Boy Player
Both the GB Micro and Nintendo DS/DS Lite lack the hardware (and the switch) to run GB/GBC games via legitimate means.
Here's what the cable itself looks like Whole cable (official) http://i791.photobucket.com/albums/yy198/gunbladelad/Game%20Boy/Linkcable.jpg
If you haven't purchased a cord yet, I can vouch for the product in the American Amazon link in the previous post. It's a little fragile, and it's not official nintendo brand, but it works perfectly. The only thing I've noticed is that it fits VERY snuggly into the GBA SP, so when you pull it out you sort of have to wiggle it back and forth carefully so that it doesn't break, but as long as you're careful it should be totally fine.
The links I gave are all for the official cables. If they're showing the official cables and not actually selling them, then they're technically violating retail laws.
It's standard practice, but it should always be stated to remove the cable by pulling the connector rather than the cable. I've seen a few link cables get trashed that way. (I still have my original Game Boy - as in the grey half-bricks - link cable working perfectly well.
Where logic won't work - I will PAL Gamers unite - http://www.gamefaqs.com/boards/277-pal-gaming