We introduce and numerically study a long-range-interaction generalization of the one-dimensional Fermi-Pasta-Ulam (FPU) \(\beta-\) model. The standard quartic interaction is generalized through a coupling constant that decays as \(1/r^\alpha\) (\(\alpha \ge 0\))(with strength characterized by \(b>0\)). In the \(\alpha \to\infty\) limit we recover the original FPU model. Through classical molecular dynamics computations we show that (i) For \(\alpha \geq 1\) the maximal Lyapunov exponent remains finite and positive for increasing number of oscillators \(N\) (thus yielding ergodicity), whereas, for \(0 \le \alpha <1\), it asymptotically decreases as \(N^{- \kappa(\alpha)}\) (consistent with violation of ergodicity); (ii) The distribution of time-averaged velocities is Maxwellian for \(\alpha\) large enough, whereas it is well approached by a \(q\)-Gaussian, with the index \(q(\alpha)\) monotonically decreasing from about 1.5 to 1 (Gaussian) when \(\alpha\) increases from zero to close to one. For \(\alpha\) small enough, the whole picture is consistent with a crossover at time \(t_c\) from \(q\)-statistics to Boltzmann-Gibbs (BG) thermostatistics. More precisely, we construct a "phase diagram" for the system in which this crossover occurs through a frontier of the form \(1/N \propto b^\delta /t_c^\gamma\) with \(\gamma >0\) and \(\delta >0\), in such a way that the \(q=1\) (\(q>1\)) behavior dominates in the \(\lim_{N \to\infty} \lim_{t \to\infty}\) ordering (\(\lim_{t \to\infty} \lim_{N \to\infty}\) ordering).