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We present measurements of the angular two-point galaxy correlation function,$w(theta)$, from the APM Galaxy Survey. The performance of various estimatorsof $w$ is assessed using simulated galaxy catalogues and analytic arguments.Several error analyses show that residual plate-to-plate errors do not bias ourestimates of $w$ by more than $10^{-3}$. Direct comparison between ourphotometry and external CCD photometry of over 13,000 galaxies from the LasCampanas Deep Redshift Survey shows that the rms error in the APM plate zeropoints lies in the range 0.04-0.05 magnitudes, in agreement with our previousestimates. We estimate the effects on $w$ of atmospheric extinction andobscuration by dust in our Galaxy and conclude that these are negligible. Weuse our best estimates of the systematic errors in the survey to calculatecorrected estimates of $w$. Deep redshift surveys are used to determine theselection function of the APM Galaxy Survey, and this is applied in Limber'sequation to compute how $w$ scales as a function of limiting magnitude. Ourestimates of $w$ are in excellent agreement with the scaling relation,providing further evidence that systematic errors in the APM survey are small.We explicitly remove large-scale structure by applying filters to the APMgalaxy maps and conclude that there is still strong evidence for moreclustering at large scales than predicted by the standard scale-invariant colddark matter (CDM) model. We compare the APM $w$ and the three dimensional powerspectrum derived by inverting $w$, with the predictions of scale-invariant CDMmodels. We show that the observations require $Gamma=Omega_0 h$ in the range0.2-0.3 and are incompatible with the value $Gamma=0.5$ of the standard CDMmodel.Comment: 102 pages, plain TeX plus 41 postscript figures. Submitted to MNRA

The APM Galaxy Survey -- III. An analysis of systematic errors in the angular correlation function and cosmological implications