duo core 478socket

I'm going to tell you what I would look at if I were going to try this. If you want to do this you might waste a few months of your spare time. if it works you will have some knowledge that others won't posess.
1- Does you MB support 64 bit Pentium 4 CPUs? All C2D chips are 64 bit. Some P4 were 32 bit.
2- What FSB does your MB support? 800, or 1066 would be good numbers (200, or 266fsb are the same).
CPU Mobile CPUs can be found at Starmicronics. There are even C2X, and C2Q cpus there but too expensive for an experiment like this.
1- The CPU must be 65nm. 45nm C2D will have too low voltage.
2- The FSB must match something that your MB supported.
BIOS issues.
1- BIOS splicing- Wimsbios and a few others might be able to splice code from a laptop BIOS into a desktop BIOS for you. No guarantee they will do this or that it will work.
2- DIY method Go to delidded.com and look into their Microcode articles. It's an LGA771 to 775 site. the microcode for your new CPU comes from Intels website. This is used to add LGA771 code to LGA775 BIOS files. This might work for Laptop 478 to DT 478.
3- Be sure you have the latest BIOS update. Things like PCIe version and memory speed and capacity can get updated there.
4- Olde MB can require older memory. Low density modules have chips on both sides, this is usually the default for old systems.
Any C2D will equal the fastest P4, at least in LGA775. But LGA775 is borderline obsolete. The best CPUs with a good O/C and modern GPU can still hang in there.
If you get this to work the P4s had high clock speeds, and high voltages so a serious O/C would then be possible! The E520 in my sig. supports P4, and 65nm C2X. I've had it up to 4GHz.
 
I don't like to tell people something won't work without giving a reason. I've been a victim of this many times my self, only to find out later that it could be done.
The gorilla in the room here is the chipset. They're hard wired and cant' be updated. The P4 series used Netburst technology which processed data in a single thread, and relied on faster and faster clock speeds for improvements. It ran into a wall at 3.73GHz. Core technology used a parallel aproach and 1.8Ghz C2D did just as much computing as the twice as fast 3.73 P4. So it had room to grow, and allowed quad cores also.
Lga775 had already happened by then. Some LGA775 chipsets are P4 only, some like the 965 in my Dell E520 accept both.
It's very unlikely that the very different instruction sets for the C2D will exist in a Socket 478 chipset.
However chipset specs. are a very funny thing. Not everything happens at once in the computer industry. You need a matching chipset, CPU, and motherboard to be available to test or sell anything. So if a new CPU is planned with maybe a faster FSB, or different on die cache setup. These features will be written into the chipset before the CPU is announced. If the CPU is delayed, then the spec. sheet for the chipset wont list support for the non existent CPU., fsb, whatever.
An example of this is the X38 chipset. Intel was planing to go to 400fsb in LGA775. It mostly got released in LGA771 Xeons. When Intel finally produced a couple 400fsb LGA775 chips for testing there was no LGA775 MB that supported them. Intel wanted them written up in the magazines and knew the MB were coming so to get the reviews written in time they revealed that the existing X38 chipset supports 400fsb.
My E520 officially supports up to Q6700@2.66 but when I plugged in a QX6800 it was recognized as a Q6800. A CPU that was never produced, yet support for it was there.
That was my first O/C on the locked BIOS Dell. It took me several months to get the unlocked multiplier going, and a couple more to get to 4GHz.
There have been successful mobile CPU retrofits in the past. The AMD K6-3+ 400 was one of them. When AMD and Intel went to Slot type CPU configuration to get L2 cache off of the MB (P2, Athlon) the mobile computer Mfg. didn't want to scrap their proprietary motherboards yet. So AMD made a Socket 7 CPU with Athlon sized lithography and on-chip L2 cache like the slotted CPUs. It was never sold for desktop use. A few escaped (mostly in Europe) some people considered them to be a myth. When installed in a Super 7 MB (100fsb = Super) they provided tri-level cache, and had jumper settings for a 6x multiplier to run 400MHz on a regular socket 7 board (4x100, or 6x66). With secret lower (than other socket7) voltage jumper settings they could be O/C 50% and ran clock for clock not just with the next gen P2/ Athlon, but with the back to socket Pentium3. Intel never shared a socket with AMD again!
It's very unlikely your idea will work. Not because it's a bad idea. I just think that even though the socket continued under another name in laptops, They had C2D chipsets to match.