The measurement of G, the Newtonian constant of Gravitation, is a long standing problem which continues to provide a challenge to our scientific creativity. Today's nearly 1% discrepancy between the recent PTB (Physikalisch-Technische Bundesanstalt) measurement and the `accepted` value of G is of considerable interest to experimental physics and the world of standards. The fact that the PTB measurement appears to be competently and thoughtfully done makes the discrepancy even more intriguing. A new free-fall approach (using g to measure G) will be used. In this experiment, an FG5 absolute gravimeter is to be employed in connection with a movable 1200 lb. Tungsten mass that surrounds the dropping chamber. Little g is measured when the surrounding mass is located above the FG5's dropping region and then when it is located below the dropping region. The 80 micro-gal difference in these two measurements together with the geometry of the surrounding mass can be used to determine G. A preliminary (proof-of- concept) experiment using a 200 lb. Mass has already been carried out. This follow-on experiment is expected to result in an accuracy for G of better than 5 parts in 10,000.