05:01.0 Network controller: Broadcom Corporation BCM4306 802.11b/g Wireless LAN Controller (rev 03)
Those are the important parts - the driver you need is for Broadcom card BCM4306. On Linux that says "bcm43xx", which usually means a little more work than with some cards. There's no need to remove it from your machine during the setup; that headache is only for Windows users.
There are possibly two ways to get it working, if it's not working out of the box. First let's see if that's the case;
sudo iwlist wlan0 scan
If that tells "interface doesn't support scanning", you can re-try it with eth1 instead of wlan0 (eth0 would be your wired network card), but usually it means the driver isn't functioning. If it is, the above command gives output; either list of networks that it "sees", or a message telling you there are no networks in the area. If it tells you there are no networks (but doesn't say the interface doesn't support scanning), the problem is elsewhere; the interface error means you need a driver.
For a driver there are probably two choices; native bcm43xx module (or something alike) and FWcutter, or NDISwrapper. Either way you need to obtain the right Windows driver (a .inf and a .sys file) for that BCM4306. Then either use FWcutter, if it works with this particular card, to chop something out of the driver and follow the FWcutter instructions to have bcm43xx module work with your card, or install ndiswrapper and ndiswrapper-utils, then use ndiswrapper to install the Windows driver (there are lots of howtos around this site and more on the web). If you use ndiswrapper, you also need to blacklist the bcm43xx kernel module to make sure it won't interfere. After this you're set up; if you plan on using WPA encryption, wpa_supplicant is probably needed (setting it up might sound tricky, but it's just as easy as using ndiswrapper, which in turn is easier than learning to play solitaire).
Maybe you're willing to search for ndiswrapper + bcm4306 - the howtos for ndiswrapper should work no matter which Linux version they're written for, as the program is the same.