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We report measurements of the magnetic penetration depth lambda(T) in high-quality CePt3Si samples down to 0.049 K. We observe a linear temperature dependence below T approximately equal to 0.16Tc, which is interpreted as evidence for line nodes in the energy gap of the low-temperature phase of this material. A kink in lambda(T) at about 0.53 K may be associated with the second superconducting transition recently reported. The results are discussed in terms of the symmetry of the superconducting order parameter.