It may or may not work. You have one of the early PCI Express 1.0 boards, and some of them have been known to have issues with PCI Express 2.1 cards. The latest BIOS revision is from 2007, so there is no BIOS update to fix this if there is a problem. You can try it, but you may have to RMA the card and look for another, as there is no guarantee that it will work.
You can try looking for a PCI Express 2.0 card which will work with your board, but you will have to look for older cards ie. Radeon HD 4000 series or earlier or Nvidia GeForce 200 series or earlier. All current cards use PCI Express 2.1.
Be aware that if you are looking for a computer with gaming capabilities, you probably should consider a new system altogether. The Pentium 4 falls below minimum system requirements for most current games. Your board only supports Pentium 4s and Celeron Ds leaving you with no real CPU upgrade options.
There are two versions of the HD 5670. A PCI-e 2.0 and PCI 2.1 version. The PCI-e 2.0 version is guaranteed to work in a PCI-e 1.0 slot, but there have been a few cases where the PCI-e 2.1 version has compatibility issues with a PCI-e 1.0 slot.