The $\pi -\pi$ amplitude in the low-energy region is parametrized in a crossing-symmetric way as the sum of the $\rho$ and $f^0$ resonance poles plus a polynomial background. The parametrization is flexible and capable of producing amplitudes having quite different features in the energy region below 1 GeV. The parameters are then varied so as to minimize the deviation from elastic unitarity on a set of closely spaced points. In addition, negative-moment finite-energy sum rules are used to connect the low-energy region with assumed Regge asymptotic behavior in the $I=1, 2$ amplitudes. With the mass and width of the $f^0$, the mass of the $\rho$, and the slope of the $\rho$ trajectory fixed, an approximate solution satisfying the constraints is found, yielding a $\rho$ width of 80± 30 MeV. The solution displays the usual characteristics of the low-energy $\pi -\pi$ amplitudes suggested by other analyses, namely, small scattering lengths and a large $I=0\mathrm{}$ $S$-wave phase shift near the mass of the $\rho$. This resonantlike behavior is found without introducing an $S$-wave pole in the parametrization, while the small scattering lengths are obtained as results of the numerical bootstrap, although no current-algebra constraints are included. Our proposed solution is also found to satisfy various inequalities proposed by Martin for the ${\pi }^0-{\pi }^0$ scattering amplitude.

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