Is it required by code to run the ground wire in the same conduit as the hot and nuetral conductors?

Yes it is. You cannot run the ground wire outside the raceway with few exceptions. NEC 300.3(B)

One exception is that you can do this if the wire is actually an equipment bonding jumper and is less than 6' long. NEC 250.102(E)

The other is in the case of grounding old ungrounded receptacles. NEC 250.130(C)