What is the weight of a single 7.62�51mm NATO armor piercing round?

That depends on which AP round in this caliber you mean since there are many options. If we take the American M61 as a baseline the bullet weighs 9.75 grams. The M993 (designed by FFV in Sweden) has a 8.3 gram bullet with a subsequent increase in muzzle velocity from 838 m/s to 930 m/s.