The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable τ₁^b in neutral-current deep-inelastic electron-proton scattering (DIS). The observable τ₁^b is equivalent to a thrust observable defined in the Breit frame. The data sample was collected at the HERA ep collider in the years 2003–2007 with center-of-mass energy of s=319GeV, corresponding to an integrated luminosity of 351.1 pb^-1. Triple differential cross sections are provided as a function of τ₁^b, event virtuality Q², and inelasticity y, in the kinematic region Q²>150GeV². Single differential cross sections are provided as a function of τ₁^b in a limited kinematic range. Double differential cross sections are measured, in contrast, integrated over τ₁^b and represent the inclusive neutral-current DIS cross section measured as a function of Q² and y. The data are compared to a variety of predictions and include long-standing and more recent Monte Carlo event generators, predictions in fixed-order perturbative QCD where calculations up to O(α_s³) are available for τ₁^b or inclusive DIS, and resummed predictions at next-to-leading logarithmic accuracy matched to fixed order predictions at O(α_s²). These comparisons reveal sensitivity of the 1-jettiness observable to QCD parton shower and resummation effects, as well as the modeling of hadronization and fragmentation. Within their range of validity, the fixed-order predictions provide a good description of the data. Monte Carlo event generators are predictive over the full measured range and hence their underlying models and parameters can be constrained by comparing to the presented data.