The board in that model has a 333mhz front side bus. If the new memory works it will run at the PC2700 333mhz speed not 400mhz. When going to buy memory for any system the first thing to look at is what is supported. Try it alone and if it works at the 333 speed you can then buy a second for seeing a 1gb total. or if unopened return it for a 512 of PC2700 memory.
It also depends on whether the old system mandates single sided dimms while you may have bought a 512 with IC chips on both sides namely dual sided. For many older boards dual sided memory simply won't work at all. Otherwise like 2048Megabytes pointed out it will run slower to match the memory standard of DDR333 for that model board. DDR333=333mhz(166mhz fsb setting seen in bios x 2 =333mhz)
When I install the new memory, how do I tell what speed it is running at? The specs I have looked at on the HP web site for my motherboard say it has a processor front side bus of 800/533/400mhz. Is this not the same thing?
The memory standard is separate from the actual cpu bus speed. PC2700 DDR333 memory is the fastest speed of memory while the actual fsb is 800mhz = 400 x2 when in dual channel mode. Intel was about cpu clock speeds at that time with AMD gaining the edge seeing more work done per cpu clock cycle rather the clock speed.
That's why you would see 200mhz per channel in the bios x 2 = 400mhz fsb there on the old Socket A boards. When a dimm is installed on the dimm slot for each channel with two dimms involved rather then two on the second slot of the same channel that then enabled the dual channel mode.
But on many older boards filling all slots forced the single channel mode instead. You simply have to go by the manufacturer's instructions for installing memory on any make or model board.
On the older Intel boards the memory standard saw a multiplier added for a higher total speed for the cpu bus speed. Prior to the Core 2s Intel had focused on clock speeds over work per clock cycle giving AMD the gaming edge.