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The IAU decides.

After multiple revisions, they establish the three following criteria:

"A Planet...

 

  1. "is a celestial body that is in orbit around the Sun."

    2.  

  2. "has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape."

    3.  

  3. "has cleared the neighbourhood around its orbit."

 

Objects meeting 1 and 2 but not 3 are now "dwarf planets"...which are in fact not planets at all.