Note: I did this for a friend's Dell Studio 1435 Laptop, while installing the latest LinuxMint 17 Qiana Mate Edition. This should work on Ubuntu / LinuxMint for any machine which has a Broadcom BCM4312 wifi (802.11b/g) adapter.
You can make sure that you have the correct device mentioned here, by typing in the following command in the Terminal:
lspci | grep -i bcm4312
It should show something like this: