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The unloaded Q's of pure Nb cavities in the TE011 mode at 11.2 GHz and in the TM010 mode at 8.4 GHz were measured between 1.2° and 4.2°K to determine whether the residual surface losses in Nb could be made sufficiently low to make it a useful material for practical microwave devices. Three forms of reactor grade Nb were evaluated: recrystallized fully wrought, arc melted, and electron-beam melted. Each of these materials was subjected to additional metallurgical processing including high-temperature vacuum firing and polishing to determine their effects on surface losses. In the TE011 mode, a residual Q of 3.8×1010 was achieved. Also in the TE011 mode, the unloaded Q was observed to be essentially a constant up to the ac critical magnetic field Hcae, where it abruptly decreased by a factor of 100 or more. An Hcae as large as 436 Oe was observed for unstrained Nb. In the TM010 mode, which has electric fields terminating on the surface of the Nb, a residual Q of 2×109 was achieved, the limitation on the maximum field level in the cavity was Hcae, and no power loss associated with electric-field emission was observed up to an electric field of 2.1×105 V/cm, which was the maximum field level obtained in the cavity. The Mattis and Bardeen theoretical expression for the superconducting surface resistance fits the data very well, and the value of the energy gap 2Δ(0) obtained from the fitting procedure is (3.72±0.01) kBTc, which agrees with values of 2Δ(0) obtained by other methods. The surface resistance data are not inconsistent with a double energy-gap model, which has been used to explain specific heat data; however, it was not possible using the data to distinguish between a double energy-gap model and a single energy-gap model with a residual surface resistance. It would be expected that Hcae would be about Hc1 where magnetic flux starts penetrating a type-II superconductor; however, the largest measured Hcae (436 Oe) was in fact about Hc1/4.