Generally one can use a replacement motherboard and still use the Hard Drive with Windows Activated on it as long as the Chipset is the same,the on board Integrated Graphics or Discrete Graphics card is the same AND the
Network Chip on board the motherboard is the same exact type.
You will probably lose a vote for Product activation if the MAC (Media Access Controller) address is different on the ethernet port.Not a major problem though but be aware that if you make 2 more deciding factor hardware changes
http://www.aumha.org/win5/a/wpa.php
you might have to reactivate windows which can be a pain in the ass (and may or may not work if you make too many hardware changes).
Myself I have previously in the past had a failure with an emachines model T6420 where the bad power supply also took out the motherboard.I found a replacement motherboard that had the same chipset,on board IGP and the same EXACT network chip but was of a more modern socket type (AM2+).The T6420 used a socket 939 motherboard.In a way the hardware failure gave me the opportunity to upgrade to a much faster dual core CPU (Athlon II X2 240) over the previous Athlon 64 3400+ CPU.
Here is your motherboard.
ECS MCP61SM-GM Mainboard
or
http://www.google.com/search?sourceid=chrome&ie=UTF-8&q=ECS+MCP61SM-GM+Mainboard
Where you could search to replace the exact board if you want the same motherboard.
Here is a great forum also for help (Unofficial eMachines Forums)
http://www.dexplor.com/phpBB2/index.php
I did see one board (and also from ECS too) at newegg which used the nForce 4 chipsets (NVIDIA MCP61S uses the NVIDIA nForce 430 SouthBridge chip) AND Geforce 6100 (close enough to 6150 also) IGP AND the same exact Network Chip (Broadcom AC131).It's a AM2+ motherboard so it will work with your AM2 CPU and you can also reuse your DDR2 type memory for it.
It's also very inexpensive.It looks like you are in perfect luck here with the following motherboard as a alternative replacement board.
http://www.newegg.com/Product/Product.aspx?Item=N82E16813135246
This looks like the alternative replacement motherboard that you will want.
It's also much less expensive than purchasing a direct replacement motherboard (same exact model) too.
The main key here again is that the chipset is the same (It is),the IGP is the same (It is) and the Network Chip is the same exact model (It is).You must also reuse all your previous components too (same amount of memory as when Windows was activated,same Optical drive,same Hard Drive etc).
Now looking at your machines specifications
http://reviews.cnet.com/desktops/emachines-t3626/4507-3118_7-32655760.html
this has a older and meager now single core AMD Sempron 3800+ / 2.2 GHz CPU.
What you could do is replace this (upgrade) with a much more powerful AM3 CPU later on if you got that ECS motherboard that I mentioned.Here is the list of CPU's that it supports.
http://www.ecs.com.tw/ECSWebSite/Product/Product_Detail.aspx?DetailID=988&CategoryID=1&DetailName=Feature&MenuID=20&LanID=9
You will lose a second vote on Windows Product Activation by upgrading the CPU (not a problem really).
You could make one more major hardware change in addition without triggering reactivation like either a memory upgrade or using a discrete graphics card but not both.
For the inexpensive price of that motherboard a Athlon II X2 or X3 (dual or triple core) CPU would make a fine upgrade giving your PC with perhaps 3 to 6 times it's current performance depending upon the chosen CPU.
Make sure though that you get a newer type and reliable power supply replacement too.
It looks like your in the same luck as I was with my revived emachines T6420.You should have no problem with that very inexpensive alternative replacement motherboard and a nice CPU upgrade just like I did (I had a spare Athlon II X2 240 CPU that I used in my rebuild).