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Could a planet made completely of water exist?

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In the Star Trek: Voyager episode "Thirty Days", there was a planet made completely out of water, which is the inspiration for this question. However, that planet was held together by an artificial containment field. I'd like to know if a water planet would be possible that would be held together just by normal gravitation.

To be clear what the properties of this water planet should be:

  • The planet is completely made out of water (that is, any non-water substance makes up a negligible amount of its mass). There may be a non-water atmosphere above, but there's no stone or metal center; the water goes down to the center.
  • Most or all of the water on the planet's surface is liquid (there may be polar ice caps, but they shall cover less than half of the planet).
  • The planet is held together by its own gravitation.

Is such a planet possible? And if so, could there be a plausible mechanism how it could be created naturally?

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