There's currently another answer which addresses the question that you asked, but not (I think) the question that you meant to ask:
Does a bound hydrogen atom have the same mass as a free election plus a free proton at rest, or does the binding energy $E=mc^2$ change this mass?
The answer here is that the masses are different: the bound atom is less massive than the sum of its constituents by the (mass equivalent of the) binding energy, which is $13.6\rm\,eV$.
That's a tiny mass-energy, only a few parts per billion of hydrogen atomic mass.
I don't believe the difference in mass has been measured directly, and I don't expect that it will be anytime soon, because the methods for measuring the masses of charged and neutral particles are quite different.
In nuclear physics, where the energies involved are much larger, you can have mass differences which can be measured directly. For example, the binding energy of the deuterium nucleus is about 0.1% of its mass.
You can imagine separately measuring the neutron mass, proton mass, deuteron mass, and the energy of the photon released in $\rm np\to d\gamma$ to check this directly.
However, again because of electric charge, it turns out that there is (so far) no other comparably-precise measurement of the neutron mass, so the deuterium-formation experiment is currently effectively a neutron-mass measurement.
(Note that the binding energy of the deuterium atom is still about $13.6\rm\,eV$; it's different from the hydrogen binding energy starting in the fourth or fifth significant figure, because the nuclear mass makes a little bit of a difference.)