Absolute Zero is a theoretical temperature, exactly 0° Kelvin (-459.67°F or -273.15°C), that is the bottom limit of thermal energy existence. At this temperature, all atoms of a substance will cease their motion, becoming absolutely solid. This is a theoretical limit because it has never been demonstrated. We do not have the capability of removing all motion from atoms, although scientists have demonstrated very low temperatures. There is no place in the universe where the temperature is below 2.73°K, because of the ambient background radiation that exists from the Big Bang.
The Law of Conservation of Energy states that all energy in a closed system will remain in existence. It may change its state, but it can never be destroyed. For example, while a piece of paper might burn up, the paper is not truly destroyed; instead, it leaves behind ashes and releases heat and smoke into the air. Although the potential energy of the ashes has been used, the matter itself is still energetic at an atomic level, and so the energy in the matter has not been destroyed.
To understand how these laws are compatible, we must understand that all matter is in constant motion. Everything humans perceive as solid matter is in fact composed of trillions of atoms, locked together by atomic bonds but still moving. If there were a way to disrupt the atomic bonds of a solid object, the object would cease to exist as a visible object as its component atoms scatter into space. The speed of an Electron orbiting the nucleus is enormously fast: we can theorize that if a person tried to touch an enormous atom, he would be unable to penetrate the shell of the electron orbit; the electron(s) would be under his hand no matter where he placed it. The motion of the electrons is energy, and the motion of the atoms continues that energy. If the atom is split, as in a nuclear reaction, all the matter is converted to heat energy; it is not destroyed but altered to a different form.
If an object were to be frozen to Absolute Zero, the motion of the atoms and their parts would stop. The object would be entirely still in every possible sense, but as soon as some form of heat touched it, the motion would start up again. In this case, the resting energy of the moving atoms would be nullified but not destroyed, and the potential energy of the atoms would remain. Even so, the matter itself would still exist in some form; if the cooling to Absolute Zero destroyed the object to atomic dust, the atoms themselves would still exist and likely release energy in the conversion from solid to atomic.
Simply put, any matter or energy continues to exist in one form or another. Cooling to Absolute Zero would only remove thermal energy by stilling the energetic motion of the atoms; the matter, and therefore potential energy, would remain.