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Spanning a broad range of physical systems, complex symmetry breaking is widely recognized as a hallmark of competing interactions. This is exemplified in superfluid ^3 He which has multiple thermodynamic phases with spin and orbital quantum numbers S = 1 and L = 1, that emerge on cooling from a nearly ferromagnetic Fermi liquid. The heavy fermion compound UPt _3 exhibits similar behavior clearly manifest in its multiple superconducting phases. However, consensus as to its order parameter symmetry has remained elusive. Our small angle neutron scattering measurements indicate a linear temperature dependence of the London penetration depth characteristic of nodal structure of the order parameter. Our theoretical analysis is consistent with assignment of its symmetry to an L = 3 odd parity state for which one of the three thermodynamic phases in non-zero magnetic field is chiral.