Can an IFR clearance be issued and flown through IMC in class G airspace?
The key point is in your last sentence: 


  In the case of class G airspace from the ground, the airspace is uncontrolled.


Uncontrolled airspace is, well, uncontrolled. An appropriately rated, current pilot, in an appropriately equipped aircraft, may fly IFR in class G airspace without either a clearance or a flight plan. There are no clearances to waive VFR minima in class G, because the flight is conducted under IFR.

If an entire flight is flown in class G airspace, a pilot does not need to talk to a controller at all. When class G airspace is overlaid by controlled airspace, like class E, the pilot will need a clearance before entering controlled airspace. The controller will provide a clearance (FAA Order  7110.65, p. 4-3-2):


  WHEN ENTERING CONTROLLED AIRSPACE (instruction), FLY HEADING (degrees) UNTIL REACHING (altitude, point, or fix) BEFORE PROCEEDING ON COURSE.


Once the pilot is released for departure, they would depart IFR in class G, where they would be responsible for their own terrain and traffic clearance (using tools like an Obstacle Departure Procedure). As soon as they climb into class E airspace, they receive the benefits and responsibilities of flying in the ATC system.

An interesting border case is raised by wbeard52. In this case, a pilot departed a class G airport with class E starting at 700 AGL, and broke out of the clouds before entering controlled airspace. The ruling basically said that taking off in this situation without a clearance is not explicitly illegal, but falls under "careless or reckless operation" prohibited by FAR 91.13.