The three-body problem is arguably the oldest open question in astrophysics and has resisted a general analytic solution for centuries. Various forms of perturbation theory provide solutions in portions of parameter space, but only where hierarchies of masses and/or separations exist. Numerical integrations show that bound, non-hierarchical triple systems of Newtonian point particles will almost always disintegrate into a single escaping star and a stable bound binary, but the chaotic nature of the three-body problem prevents the derivation of analytic formulae that deterministically map initial conditions to final outcomes. Chaos, however, also motivates the assumption of ergodicity. I will present a new statistical solution to the non-hierarchical three-body problem that is derived using the ergodic hypothesis and that provides closed-form distributions of outcomes (for example, binary orbital elements) when given the conserved integrals of motion. We compare our outcome distributions to large ensembles of numerical three-body integrations and find good agreement, so long as we restrict ourselves to "resonant" encounters (the roughly 50% of scatterings that undergo chaotic evolution). In analysing our scattering experiments, we identify "scrambles" (periods of time in which no pairwise binaries exist) as the key dynamical state that ergodicizes a non-hierarchical triple system. I will briefly discuss how the generally super-thermal distributions of survivor binary eccentricity that we predict have applications to many astrophysical scenarios. For example, non-hierarchical triple systems produced dynamically in dense star clusters are a primary formation channel for black-hole mergers, but the rates and properties of the resulting gravitational waves depend on the distribution of post-disintegration eccentricities.