A unitarized nonrelativistic meson model which is successful for the description of the heavy and light vector and pseudoscalar mesons yields, in its extension to the scalar mesons but for the same model parameters, a complete nonet below 1 GeV. In the unitarization scheme, real and virtual meson-meson decay channels are coupled to the quark-antiquark confinement channels. The flavor-dependent harmonic-oscillator confining potential itself has bound states ǫ(1.3 GeV), S(1.5 GeV), δ(1.3 GeV), κ(1.4 GeV), similar to the results of other bound-state qq models. However, the full coupled-channel equations show poles at ǫ(0.5 GeV), S(0.99 GeV), δ(0.97 GeV), κ(0.73 GeV). Not only can these pole positions be calculated in our model, but also cross sections and phase shifts in the meson-scattering channels, which are in reasonable agreement with the available data for ππ, ηπ and Kπ in S-wave scattering.