I need suggestions for a better processor
What would be my best bet? It needs to be socket 478/n, and it runs in a DDR1 motherboard with DDR1 memory, I want it to be more powerful than my current processor(2.6ghz single core) and if possible a dual core.
What is the best processor I could get of that spec?
You are probably out of luck in this case. There are no dual core CPUs on Socket 478. You may be able to get a slightly faster Pentium 4, say 2.8 or 3.0 GHz maybe, you would have to check your mobo's manufacturer's website for a CPU support list for your particular motherboard, it's possible your current CPU is the best you can get.
I would seriously consider a new build at this point, Socket 478 has been dead for a long time, and even a cheap bottom of the shelf computer that you can find at Best Buy or Wal Mart is vastly superior to that old rig. There's no sense in pouring money into technology that is extremely obsolete at this point.
As Supernova said socket 478 has been discontinued for 7 years or something like that. The socket 775 and then the release of the Intel® Core™ 2 Duo processors is just about the lowest level of processor that you are going to find with a dual core design that you would want to be using today. Even the low end of the Intel® Celeron® G550 which is $60 and a H61 chipset based board you could pick up a system that would dwarf anything in socket 478.