Here is an Arch Linux question that deals with your exact same device 14e4:43a0. https://aur.archlinux.org/packages/wireless-bcm43142-dkms/ The proposed solution is the latest version of bcmwl-kernel-source, in that case 6.30.223.30-1. The result reported later was: "THANK YOU!!! It works as expected."
In Ubuntu 13.04, there is a proposed version of bcmwl-kernel-source 6.30.223.30+bdcom-0ubuntu3. If you are running 13.04, open Software and Updates and select Prereleased Updates under the Updates tab. With a temporary ethernet connection, open a terminal and do:
sudo apt-get update
sudo apt-get install bcmwl-kernel-source
Check the version that was installed:
sudo dpkg -s bcmwl-kernel-source
It should say: Version: 6.30.223.30+bdcom-0ubuntu3
Now load the driver:
sudo modprobe wl
Is it working now? So that you don't get a lot of unstable software, go back and uncheck Prereleased Updates and re-run:
sudo apt-get update