PCI Express® based PC is required with one X16 lane graphics slot available on the motherboard
500 Watt or greater power supply with two 75W 6-pin PCI Express® power connectors recommended (600 Watt and four 6-pin connectors for AMD CrossFireX™ technology in dual mode)
Certified power supplies are recommended. Refer to
http://ati.amd.com/certifiedPSU for a list of Certified products
Minimum 1GB of system memory
Installation software requires CD-ROM drive, a keyboard, a mouse, and a display
DVD playback requires DVD drive and a DVD
Blu-ray™ playback requires Blu-ray drive and a Blu-ray disc
For an AMD CrossFireX™ system, a second AMD Radeon™ HD 6800 Series Graphics card, an AMD CrossFireX™ Ready motherboard and one AMD CrossFireX™ Bridge Interconnect cable is required
Copied directly from
http://www.amd.com/us/products/desktop/graphics/amd-radeon-hd-6000/hd-6870/Pages/amd-radeon-hd-6870-overview.aspx#3
Also newegg says Minimum of a 650 Watt power supply on the 2GB model from
http://www.newegg.com/Product/Product.aspx?Item=N82E16814129183
In short you will need a stronger PSU even for the 1GB model of the AMD HD Radeon 6870 video card