Yes, but you're pretty close.. I just put an E2000 I got on Craigslist for $10 in my garage two days ago.
Wired in as I described above, it is now my third wireless access point. Before this, the wifi signal in my detached garage sucked. Now it's awesome.
[edited below to include the correct order to do things to make your life easiest]
First:
Change old router to a different IP address (usually your internet gateway router will be 192.168.1.1, so set the barn router's inside IP address to 192.168.1.2.
Turn off DHCP on the soon to be barn router.
Change this before installing the new router, and THEN remove the old router.
Plug your internet into the internet port on the house router (not related to this, just covering all bases), replace the old router with the new one and get your internet working again.
Second.
Make sure that the DHCP range on the ONE router that is doing dhcp (which should be the new house router) does not include 192.168.1.2 (start it at a higher number, like 192.168.1.100).
Plug the wire connecting the barn to the house into the LAN port on the house router.
Plug the connecting wire into the LAN port on the barn router.
Optional: Put a piece of tape over the internet/WAN port on the barn router (you won't be using it).
You can now plug any other wired devices into the remaining LAN ports.
If you can, set both routers to the same SSID (this is case sensitive) and password (this will ONLY work if both are set to use the same security settings such as WPA2).
If the old router does not support the same type of wifi network as the new one, you should just create a new SSID (though this creates other issues).
In your case, you may have a problem though. The WRT54G supports B/G networks, while the EA3500 supports N. You can use the same SSID for all access points on a network, but only if the security is the same on all nodes. In your case, if you use WPA2 personal, it should work.
Wifi devices are stupid. If they're connected to a network, they will stay connected to that SSID until they lose their connection. And they'll connect to the first network that they recognize.
If you choose a different SSID for the house and the barn, then if your phone connects to the wifi in the house, when you walk to the barn, so long as it still has minimal signal, it will try to stay connected to the house. If that still works, great. But my issue has been that once you get too far away, you remain connected, but may not be able to use the bandwidth for squat. Unless they're so far apart that each different network cannot be seen from the opposite end, you may find that devices you carry up and back will be stuck to the wrong access point, and your signal will be awful.
If you choose the same SSID/security on all your access points, then your device will always think it is connected to the same network, and will automatically attach to the access point with the most powerful signal (automatic roaming).
Set both routers to a different channel.
USE ONLY channels 1,6, or 11, and NOTHING in between.
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