We have evaluated the $\pi \pi$ 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 $J=A+c\partial \phi$, where $A$ is the usual axial-vector current, $c$ is the Goldberger-Treiman constant, and $\phi$ 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 $\sigma$ 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 $\epsilon$ resonance is the only $s$ wave dominating the low-energy ($\pi \pi$) scattering, we find a relation connecting the form factors arising from the equal-time commutators of the current $J$ to the $\epsilon$-pion coupling constant. Finally, we obtain the scattering lengths corresponding to isospins 0 and 2, and $\epsilon$ resonance width 200 MeV, as $a_0=0.278\mathrm{}{m}^{-1\mathrm{}}$ and $a_2=-0.044\mathrm{}{m}^{-1\mathrm{}}$, where $m$ is the pion mass.

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