The primordial deuterium abundance probes fundamental physics during the big bang nucleosynthesis and can be used to infer cosmological parameters. Observationally, the abundance can be measured using absorbing clouds along the lines of sight to distant quasars. Observations of the quasar PKS1937-101 contain two absorbers for which the deuterium abundance has previously been determined. Here, we focus on the higher redshift one at z(abs) = 3.572. We present new observations with significantly increased signal-to-noise ratio that enable a far more precise and robust measurement of the deuterium to hydrogen column density ratio, resulting in DI/HI = 2.62 +/- 0.05 x 10(-5). This particular measurement is of interest because it is amongst the most precise assessments to date and it has been derived from the second lowest column-density absorber [N(H I) = 17.9 cm(-2)] that has so-far been utilized for deuterium abundance measurements. The majority of existing high-precision measurements were obtained from considerably higher column density systems [i.e. N(H I) > 19.4 cm(-2)]. This bodes well for future observations as low column density systems are more common.