I'm not 100% sure, but it should work as you expect.
First, I would make sure that both Windows and Linux kernels from your current installation support SATA drives. You can attach the SATA drive and see if you can access it from both systems.
Read grub man and find out what changes you need to make after cloning the disk. With proper config you can even try to boot Linux from your SATA drive without disconnecting the IDE drive, using already existing boot loader on the IDE drive. In fact the whole operation should be possible when you boot into your existing Linux installation, without need of live CD.
To boot Windows you probably must make your new SATA drive to have the same drive letter (C: or something), so you will have to disconnect or disable your IDE drive. (Interesting is that you mentioned your IDE drive is discovered as /dev/sdc, so it may happen, that when both disks are installed the first partition on SATA one - /dev/sda1 - becomes C: without need of any changes in BIOS settings)
For cloning, partition manipulation etc. I recommend SystemRescueCd