We present stellar evolutionary tracks and nucleosynthetic predictions for a grid of stellar models of low- and intermediate-mass asymptotic giant branch (AGB) stars at $Z=0.001$ ([Fe/H]$=-1.2$). The models cover an initial mass range from 1 M$_{odot}$ to 7 M$_{odot}$. Final surface abundances and stellar yields are calculated for all elements from hydrogen to bismuth as well as isotopes up to the iron group. We present the first study of neutron-capture nucleosynthesis in intermediate-mass AGB models, including a super-AGB model, of [Fe/H] = $-1.2$. We examine in detail a low-mass AGB model of 2 M$_{odot}$ where the $^{13}$C($alpha$,$n$)$^{16}$O reaction is the main source of neutrons. We also examine an intermediate-mass AGB model of 5 M$_{odot}$ where intershell temperatures are high enough to activate the $^{22}$Ne neutron source, which produces high neutron densities up to $sim 10^{14}$ n cm$^{-3}$. Hot bottom burning is activated in models with $M geq 3$ M$_{odot}$. With the 3 M$_{odot}$ model we investigate the effect of varying the extent in mass of the region where protons are mixed from the envelope into the intershell at the deepest extent of each third dredge-up. We compare the results of the low-mass models to three post-AGB stars with a metallicity of [Fe/H] $simeq -1.2$. The composition is a good match to the predicted neutron-capture abundances except for Pb and we confirm that the observed Pb abundances are lower than what is calculated by AGB models.
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