Actually, unofficial 6.30.223.30 Broadcom device driver works correctly for me with the same device as you (BCM4313) on Fedora 18 with 3.9.x kernels, but I didn't tried with 3.10.x kernels (professional usage of my laptop). I proposed in the bug report mentioned above a patch that makes this driver to compile for 3.10 kernel, but I can't say if it is working with such a kernel (never tested).
@Adam Porter
Maybe you should add this bug report too: https:/ /bugs.launchpad .net/ubuntu/ +source/ bcmwl/+ bug/1157880
Actually, unofficial 6.30.223.30 Broadcom device driver works correctly for me with the same device as you (BCM4313) on Fedora 18 with 3.9.x kernels, but I didn't tried with 3.10.x kernels (professional usage of my laptop). I proposed in the bug report mentioned above a patch that makes this driver to compile for 3.10 kernel, but I can't say if it is working with such a kernel (never tested).
Hope this will help you to solve this problem.
Cordially,
--
NVieville