The pp-->pp pi(+) pi(-) reaction has been measured exclusively near threshold at CELSIUS. The total cross sections are nearly an order of magnitude lower than expected from previous inclusive measurements. The differential cross sections reveal pp-->pp(*)(1440)-->pp sigma = pp(pi(+)pi(-))(I = l = 0) as the dominant process as well as significant contributions from p(*)-->Delta(++)pi(-)-->psigma. The observed anisotropy in the proton angular dependence is consistent with heavy-meson exchange. In the invariant mass spectra, no narrow structures of statistical relevance (3sigma) are found.