We have confirmed the existence, as first reported by Kim and Chan, of a supersolid state in solid 4He at temperatures below 250 mK. We have employed a torsional oscillator cell with a square cross section to insure a locking of the solid to the oscillating cell. We find that NCRI signal is not a universal property of solid 4He, but can be eliminated through an annealing of the solid helium sample. This result has important implications for the nature of the supersolid state.