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We report on the observation of nonclassical rotational inertia in solid helium-4 confined to an annular channel in a sample cell under torsional motion, demonstrating superfluid behavior. The effect shows up as a drop in the resonant oscillation period as the sample cell is cooled below 230 millikelvin. Measurement of 17 solid samples allows us to map out the boundary of this superfluid-like solid or supersolid phase from the melting line up to 66 bars. This experiment indicates that superfluid behavior is found in all three phases of matter.