The contribution of unresolved sources to the diffuse gamma-ray background could induce anisotropies in this emission on small angular scales. We analyze the angular power spectrum of the diffuse emission measured by the Fermi LAT at Galactic latitudes |b| > 30 deg in four energy bins spanning 1 to 50 GeV. At multipoles \ell \ge 155, corresponding to angular scales \lesssim 2 deg, angular power above the photon noise level is detected at >99.99% CL in the 1-2 GeV, 2-5 GeV, and 5-10 GeV energy bins, and at >99% CL at 10-50 GeV. Within each energy bin the measured angular power takes approximately the same value at all multipoles \ell \ge 155, suggesting that it originates from the contribution of one or more unclustered source populations. The amplitude of the angular power normalized to the mean intensity in each energy bin is consistent with a constant value at all energies, C_P/<I>^2 = 9.05 +/- 0.84 x 10^{-6} sr, while the energy dependence of C_P is consistent with the anisotropy arising from one or more source populations with power-law photon spectra with spectral index \Gamma_s = 2.40 +/- 0.07. We discuss the implications of the measured angular power for gamma-ray source populations that may provide a contribution to the diffuse gamma-ray background.Read less <