I ran into the same problem as you. I solved it by following the instructions on the help pages. The [instructions if you have wired internet] are the easiest - it will most probably be something like:
sudo apt-get install firmware-b43-installer
and after a reboot you'll be good to go.
I, however, didn't have a wired connection at hand (seems my ethernet is not working) and I had to follow the no-internet instructions. Basically, I first got the .deb files on another computer where I had internet:
You have to search the packages portal under your Ubuntu version. You can install them with:
sudo dpkg -i b43-fwcutter*
sudo dpkg -i firmware-b43-installer*
Now, the second one will fail, as it tries to download the latest firmware from the internet. We can copy the link that it wanted to fetch, however, and make understanding our hardware specifics and finding the appropriate version that much easier. For me the url was:
You can then get that file, copy it to the problematic machine, extract it:
tar xfvj broadcom-wl-*
and install it manually with something like:
sudo b43-fwcutter -w /lib/firmware broadcom-wl-5.100.138/linux/wl_apsta.o
(the exact details, file name, driver version, will probably differ, and you should conclude for yourself based on the help page and some common sense).
A reboot was then enough for my driver to start working, but there is some talk of potentially having to blacklist other drivers - explore this if it still doesn't work for you.