Good responses, but just to remind people, we actually still use similar methods to Lavoisier in school while learning technique and fundamental principles. It’s not like the guy did his thing and we’re all just believing him. These basic experiments have been replicated probably millions of times over the years just by students.
One way to do it would be using a mass spectrometer, which is essentially shooting an ionized atom through a magnetic field and measuring how much it is being deflected.
[Francis William Aston](https://en.wikipedia.org/wiki/Francis_William_Aston) got the Nobel price for chemistry for his work on mass spectroscopy in 1922, and there was [some work](https://en.wikipedia.org/wiki/History_of_mass_spectrometry) in that direction before that, so I’d say the answer to your second question should be yes.
Originally it was measured by people like Lavoisier doing careful experiments measuring the ratios of the weights of elements reacting with each other. For instance the weight of mercury and the weight when it has all been converted to the oxide.
Nowadays, mass spectrometry is one of the most common techniques. That depends on bending a beam of ionized material (atoms or molecules) by a magnetic or electric field. The heavier the ion, the less it deviates. Other related technology can also be used.
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