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Part of the energy is mc^2. We call this the _rest energy_.
I suppose you could define potential energy to be equal to
the rest energy, but not everybody does. Also there are
issues with the gauge-dependence of typical notions of
potential energy.
Let's be clear: The equation E=mc^2 applies in the rest
frame of the particle. This is Einstein's original meaning
of the equation, and also the meaning that has been
conventional among those who do relativity for a living, and
has been for several decades that I know of. (I am aware
that from time to time there have been attempts to define
some notion of "relativistic mass", but this is a Bad
Idea./2/) Mass (m) means rest mass, so it is redundant but
harmless to call it "rest" mass.
Bottom line: Energy is conserved. Period. There are no
known exceptions. Of course to make this work you need to
include all forms of energy, including the rest energy mc^2.