  Topological semimetals are three dimensional materials with
symmetry-protected massless bulk excitations. As a special case, Weyl
nodal-line semimetals are realized in materials either having no inversion or
broken time-reversal symmetry and feature bulk nodal lines. The 111-family of
materials, LaNiSi, LaPtSi and LaPtGe (all lacking inversion symmetry), belong
to this class. Here, by combining muon-spin rotation and relaxation with
thermodynamic measurements, we find that these materials exhibit a fully-gapped
superconducting ground state, while spontaneously breaking time-reversal
symmetry at the superconducting transition. Since time-reversal symmetry is
essential for protecting the normal-state topology, its breaking upon entering
the superconducting state should remarkably result in a topological phase
transition. By developing a minimal model for the normal-state band structure
and assuming a purely spin-triplet pairing, we show that the superconducting
properties across the family can be described accurately. Our results
demonstrate that the 111-family reported here provides an ideal test-bed for
investigating the rich interplay between the exotic properties of Weyl
nodal-line fermions and unconventional superconductivity.
