We have evaluated the $\ensuremath{\pi}\ensuremath{\pi}$ scattering lengths, using current algebra but without the use of a power-series expansion or extrapolation of the scattering amplitude. We have used the usual Lehmann-Symanzik-Zimmermann reduction in terms of an axial-vector current $J=A+c\ensuremath{\partial}\ensuremath{\varphi}$, where $A$ is the usual axial-vector current, $c$ is the Goldberger-Treiman constant, and $\ensuremath{\varphi}$ is the pion field. The scattering amplitude is decomposed into four terms, three of which are due to the equal-time commutators; two of these three are evaluated using the $\ensuremath{\sigma}$ model in order to show that the usual current algebra of the current $A$ holds for the current $J$. The other terms are evaluated in terms of single-particle intermediate states; we show that among these states, only $s$ waves contribute at threshold. Assuming that the $\ensuremath{\epsilon}$ resonance is the only $s$ wave dominating the low-energy ($\ensuremath{\pi}\ensuremath{\pi}$) scattering, we find a relation connecting the form factors arising from the equal-time commutators of the current $J$ to the $\ensuremath{\epsilon}$-pion coupling constant. Finally, we obtain the scattering lengths corresponding to isospins 0 and 2, and $\ensuremath{\epsilon}$ resonance width 200 MeV, as ${a}_{0}=0.278{m}^{\ensuremath{-}1}$ and ${a}_{2}=\ensuremath{-}0.044{m}^{\ensuremath{-}1}$, where $m$ is the pion mass.