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The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height.
In this case, the diameter of the cone is 2 cm, so the radius is equal to half the diameter, which is 1 cm.
Plugging the values into the formula, we get V = (1/3)(3.14)(1^2)(6) = 6.28 cubic centimeters.
Therefore, the correct answer is 6.28 cubic centimeters.