The issue seems to lie in whether or not the AE you have supports WDS. The WDS protocol supports bridging to the wired connection when extending (bridging) the wireless network. It probably suffices to say that newer AEs support WDS out of the box, while older ones may not (but try updating firmware first, you may have luck):
http://discussions.apple.com/thread.jspa?messageID=8292347
It would be important to note that, even if your AE supports WDS, it would downgrade the signal to 802.11 b/g when in bridged mode.
Here’s an example of a recent Slingbox support article detailing a similar setup:
http://support.slingmedia.com/get/KB-005602.html
Although it’s not Apples-to-Apples (pun intended), bridging to a Mac Mini would seem plausible (mini only has 10BaseT Ethernet).
Apparently, you can set this up using the Airport Utility; First, you need to do a hard reset on your AE; get the paper clip, hold in the reset button, while plugging in, and continue to hold until the LED blinks Amber;
Next, launch Airport Utility, then check to see the AE appear in the router column on the left; once you see it, select it, then go click on manual setup;
-Next under the Airport tab, select the wireless tab, and then under Wireless Mode, select Participate in a WDS Network; You should then be prompted to configure the AE to connect to the main (Time Capsule/Airport Extreme) network, etc. in a series of pop-up windows:
http://support.apple.com/kb/HT2044
This article describes a similar setup, only using Airport Base stations…
Lastly, there is a way to find out if your AE supports WDS mode, if you click on this link:
http://support.apple.com/kb/HT2044?viewlocale=en_US
Lastly, some additional tips regarding testing the IPv6…as many of the posts indicate (and robmandu), it’s definitely in the “black arts” realm of networking. Worse comes to worse, get a USB wifi adapter and connect to the mini and you’re set…good luck!
http://tech.kateva.org/2005/10/fixing-broken-wds-wired-to-wireless.html