I seem to recall postings of similar issues in these forums a couple of years ago, but I think the later drivers took care of those issues. Also, with the built-in connections, using the latest BIOS might be important too. The driver version you listed is old, so I would check to see if HP has a newer driver posted for your model. I think you are usually better off using whatever drivers the manufacturer supports. However, if HP does not have a newer driver for that model, you can always try the latest package from Intel.
http://downloadcenter.intel.com/Detail_Desc.aspx?agr=Y&DwnldID=18713 has the latest generic drivers for the "Intel(R) 82567LM-3 Gigabit Network Connection" dated 07/20/2011, version 220.127.116.11.
I hope this helps.