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 at band power to be less than 105 µK² 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 σ₈ = 0:8 and masking bright sources, the best fit value for the remaining power is Cℓ = 1.1⁺⁰·⁹₋₀.₈ x10⁻⁵µK² (1.7⁺¹·⁴₋₁.₃ Jy² sr⁻¹). This agrees well with model predctions 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 uxes to be Sv ∝ v²·⁶⁺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 uctuation amplitude of σ₈ < 1.18 at 95% confidence.