Obtain another desktop, hook the HDD into a free SATA port and external to a USB port. Copy everything over as normal.
If you have a laptop or otherwise cant access the guts of the machine, get something like this.http://www.newegg.com/Product/Product.aspx?Item=N82E16812232002
Hook up your desktop HDD to it and plug into the laptop via USB, as well as your external HDD. They should both appear as external drives to the machine, transfer as normal.