  We present measurements of the angular power spectrum of millimeter wave
anisotropies with the APEX-SZ instrument. APEX-SZ has mapped 0.8 square degrees
of sky at a frequency of 150 GHz with an angular resolution of 1'. These new
measurements significantly improve the constraints on anisotropy power at 150
GHz over the range of angular multipoles 3000 < l < 10,000, limiting the total
astronomical signal in a flat band power to be less than 105 microK^2 at 95%
CL. We expect both submillimeter-bright, dusty galaxies and to a lesser extent
secondary CMB anisotropies from the Sunyaev-Zel'dovich effect (SZE) to
significantly contribute to the observed power. Subtracting the SZE power
spectrum expected for sigma_8=0.8 and masking bright sources, the best fit
value for the remaining power is C_l = 1.1^{+0.9}_{-0.8} x 10^{-5} micro K^2
(1.7^{+1.4}_{-1.3} Jy^2 sr^{-1}). This agrees well with model predictions for
power due to submillimeter-bright, dusty galaxies. Comparing this power to the
power detected by BLAST at 600 GHz, we find the frequency dependence of the
source fluxes to be S_nu ~ nu^{2.6^{+0.4}_{-0.2}} if both experiments measure
the same population of sources. Simultaneously fitting for the amplitude of the
SZE power spectrum and a Poisson distributed point source population, we place
an upper limit on the matter fluctuation amplitude of sigma_8 < 1.18 at 95%
confidence.
