Gravity, matter vs antimatter

This is a perfectly good argument and one of the reasons that all the physicists I know believe that antimatter behaves just like matter in a gravitational field.

It is important to distinguish between antimatter, which is well understood from countless collider experiments, and negative matter (also known as exotic matter), which has never been observed. Antimatter does not have a negative mass. Indeed antimatter is just perfectly ordinary matter - we think it special only because we are made from matter and therefore biased. Negative/exotic matter is very different. If it existed it would cause all sorts of problems with conservation of energy and the stability of the universe.

In addition to John's answer:

There is a subtlety in antimatter. In the standard model it is axiomatic that matter and antimatter have the same sign mass. But as long as gravity is not quantized in a theory of all four forces, it is possible that antiparticles,even having a positive mass, instead of being attracted gravitationally by particles, are repulsed.

An experiment is running at CERN to check the assumption that antiparticles fall under the force of gravity.

informed in a comment that two more experiments are running at CERN to determine the behavior of antimatter to gravity: aegis and gbar.

The short answer is that we don't know. While we have made some small amounts of anti-matter, it has not lasted long enough to measure the very weak force of gravity.

However, people have speculated. Here is a long article on Wikipedia about that, with arguments going both ways.

The introduction to that article states

While the consensus among physicists is that gravity will attract both matter and antimatter at the same rate that matter attracts matter, there is a strong desire to confirm this experimentally

The thought experiment you describe is one of the arguments in favor of this consensus. (look for Phillip Morrison in the article) To my layman's mind it is a convincing argument, but we never really know until we have measured it.