The superconducting gap structure of recently discovered heavy fermion CePt(3)Si without spatial inversion symmetry was investigated by thermal transport measurements down to 40 mK. In zero field a residual T-linear term was clearly resolved as T --> 0, with a magnitude in good agreement with the value expected for a residual normal fluid with a nodal gap structure, together with a T2 dependence at high temperatures. With an applied magnetic field, the thermal conductivity grows rapidly, in dramatic contrast to fully gapped superconductors, and exhibits one-parameter scaling with T/sqrt[H]. These results place an important constraint on the order parameter symmetry; that is, CePt(3)Si is most likely to have line nodes.