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This item, when attached to a Pokémon, halves the Pokémon's weight.
Yeah... no. The only possible reason you would want to hold this item is to reduce damage from Grass Knot and Low Kick. Most of the Pokemon who would be worried about getting hit by either of these moves are Rock- or Steel-types, most of whom are actually so heavy that this item makes little difference regardless. This is, of course, ignoring the fact that the power boosts to Giga Drain and Drain Punch in BW have meant that Grass Knot and Low Kick have been almost entirely pushed out of usability for most Pokemon who use these moves. You will get very little reward from Float Stone that you wouldn't get from a more conventional item, such as Life Orb or Choice Band.
Opelucid City - Talk to Linebacker in Drayden/Iris's house.