Ive done this many times with old (40 to 240 meg, yup meg) ide drives, here are a few tricks Ive found that may help.
Make sure its on its own channel, temporarily disconnect anything else on that cable if you have a anything on it. Really old drives dont mix well with newer ones, and it may not work at all if another drive is on that cable even if jumered right. if its really old you may have to manually set the CHS settings in the BIOS (copy them from the old comps BIOS screen or get that info from the manufacturers site if its not printed on the drive label) as some really old ide drives didnt auto report that. use pio mode 2 or 3 or 4, dont use dma, some old drives get flakey under dma. set the old hd as master, master with no slave, or no jumper.. some old drives had weird jumper settings so try it in that order. Plug it into the last connector on the cable. watch the orientation of the cable, old drives werent keyed and can be plugged in "backwards."
BTW almost all inexpensive nics will have dos drivers, jyst check the web sites for the card. biggest concern is what kind of slots does the old comp use? Might not have PCI slots, only ISA, in which case youre out of luck and the only way to xfer is to put the old hd in another comp, or get a null modem adapter (crossover box) and use serial cables. Or even set it up and xfer via modem if you have 2 phone lines, or set it up at a friends house and use modems.