The BCM4313 broadcom wireless card is compatible with the open source brcm80211 driver directly included in the standard kernel or the proprietary broadcom-wl driver that can be installed from the Additional Drivers in Ubuntu. At any point of time, you should only have one of the drivers installed and trying to use it simultaneously will only result in the drivers conflicting each other.
It should be also noted that broadcom-wl has been causing more problems than resolving them and you should probably avoid them altogether if possible.
To answer your problem, I have used the steps provided in this forums thread. This method is more like a workaround for the problem rather than a permanent fix to the issue.
First of all, blacklist the wl, brcm80211 and b43 modules and all its dependencies.
gksudo gedit /etc/modprobe.d/blacklist.conf
After opening the file, add the following to it:
And then, edit the rc.local file:
gksudo gedit /etc/rc.local
And add the following above exit 0:
This will ensure that the brcm80211 driver module is loaded at the end of the boot process. You might need to call
update-initramfs -u to ensure that your changes will be effective after a reboot.
I should probably add the fact that I have BCM4313 wireless driver in my system and I have used these procedures to get it working properly.