We present 30 GHz measurements of the angular power spectrum of the cosmic microwave background ( CMB ) obtained with the Sunyaev-Zel ’ dovich Array . The measurements are sensitive to arcminute angular scales , where secondary anisotropy from the Sunyaev-Zel ’ dovich effect ( SZE ) is expected to dominate . For a broad bin centered at multipole 4066 we find 67 ^ { +77 } _ { -50 } ~ { } { \mu { K } ^ { 2 } } , of which 26 \pm 5 ~ { } { \mu { K } ^ { 2 } } is the expected contribution from primary CMB anisotropy and 80 \pm 54 ~ { } { \mu { K } ^ { 2 } } is the expected contribution from undetected radio sources . These results imply an upper limit of 155 { \mu { K } ^ { 2 } } ( 95 % CL ) on the secondary contribution to the anisotropy in our maps . This level of SZE anisotropy power is consistent with expectations based on recent determinations of the normalization of the matter power spectrum , i.e. , \sigma _ { 8 } \sim 0.8 .