The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable $\tau_1^b$ in neutral-current deep-inelastic electron-proton scattering (DIS) at the HERA $ep$ collider. The data, collected from 2003 to 2007 with a center-of-mass energy of $\sqrt{s} = 319$ GeV and an integrated luminosity of 351.1 pb$^{-1}$, are used to measure triple differential cross sections as a function of $\tau_1^b$, event virtuality $Q^2$, and inelasticity $y$. Single and double differential cross sections are also provided. The data are compared to various theoretical predictions, including classical and modern Monte Carlo event generators, fixed-order perturbative QCD calculations up to $\mathcal{O}(\alpha_s^3)$, and resummed predictions at next-to-leading logarithmic accuracy. The comparisons reveal sensitivity to QCD parton showering, resummation, and hadronization effects. Fixed-order predictions provide a good description of the data within their range of validity, while Monte Carlo event generators are predictive over the full measured range, allowing for constraints on their underlying models and parameters.The H1 Collaboration reports the first measurement of the 1-jettiness event shape observable $\tau_1^b$ in neutral-current deep-inelastic electron-proton scattering (DIS) at the HERA $ep$ collider. The data, collected from 2003 to 2007 with a center-of-mass energy of $\sqrt{s} = 319$ GeV and an integrated luminosity of 351.1 pb$^{-1}$, are used to measure triple differential cross sections as a function of $\tau_1^b$, event virtuality $Q^2$, and inelasticity $y$. Single and double differential cross sections are also provided. The data are compared to various theoretical predictions, including classical and modern Monte Carlo event generators, fixed-order perturbative QCD calculations up to $\mathcal{O}(\alpha_s^3)$, and resummed predictions at next-to-leading logarithmic accuracy. The comparisons reveal sensitivity to QCD parton showering, resummation, and hadronization effects. Fixed-order predictions provide a good description of the data within their range of validity, while Monte Carlo event generators are predictive over the full measured range, allowing for constraints on their underlying models and parameters.