The Ne22(p,γ)Na23 reaction, part of the neon-sodium cycle of hydrogen burning, may explain the observed anticorrelation between sodium and oxygen abundances in globular cluster stars. Its rate is controlled by a number of low-energy resonances and a slowly varying nonresonant component. Three new resonances at Ep=156.2, 189.5, and 259.7 keV have recently been observed and confirmed. However, significant uncertainty on the reaction rate remains due to the nonresonant process and to two suggested resonances at Ep=71 and 105 keV. Here, new Ne22(p,γ)Na23 data with high statistics and low background are reported. Stringent upper limits of 6×10-11 and 7×10-11 eV (90% confidence level), respectively, are placed on the two suggested resonances. In addition, the off-resonant S factor has been measured at unprecedented low energy, constraining the contributions from a subthreshold resonance and the direct capture process. As a result, at a temperature of 0.1 GK the error bar of the Ne22(p,γ)Na23 rate is now reduced by 3 orders of magnitude.