How do we determine mass in a vacuum?

Vacuum doesnt imply there is no gravity

In zero g, you can determine mass by measuring inertia. For example, put the mass on a spring and measure the period of oscillation.

Mass can be determined either by weight or by inertia. Weight is how much force gravity exerts on it. Inertia is how much force is needed to accelerate it a given amount. Neither of these are affected by vacuum. In fact, it is easier to determine weight in a vacuum, because you don't have to worry about taking buoyancy in air (how much it "floats") into account. If you meant a zero gravity or freefall situation instead of a vacuum, then the inertia is still exactly the same and you can determine mass that way.

How do we determine mass in a vacuum?

 

That is easy! m=0. A vacuum is a space with nothing in it and nothingness has no mass.

A practical relalisation of putting it an a spring and measurng the peiod of oscillation is given here.

http://en.wikipedia.org/wiki/Quartz_crystal_microbalance

These things are used in vacuo as often as not.