I'm not a physicist, but it's irritating to me that this article is using weight and mass interchangeably. I doubt they'd ever allow this, but the "simple" solution is to bring the weight aboard the ISS and capture its mass on an inertial balance.
Also, the ISS is a tricky place to work for a precision measurement. It's electrically, seismically, and gravitationally noisy (and huge gradients). Precision gravitational measurements are generally carried out in dedicated spacecraft with careful attention to those concerns, if they can't be done on the ground.
Some nice discussion: http://physics.stackexchange.com/questions/29929/gravity-on-...
The article doesn't mention mass at all and it is true that if a stationary object is losing mass it is also losing weight.