  A preponderance of evidence suggests that the ground state of the
nearest-neighbor $S = 1/2$ antiferromagnetic Heisenberg model on the kagome
lattice is a gapless spin liquid. Many candidate materials for the realization
of this model possess in addition a Dzyaloshinskii-Moriya (DM) interaction. We
study this system by tensor-network methods and deduce that a weak but finite
DM interaction is required to destabilize the gapless spin-liquid state. The
critical magnitude, $D_c/J \simeq 0.012(2)$, lies well below the DM strength
proposed in the kagome material herbertsmithite, indicating a need to reassess
the apparent spin-liquid behavior reported in this system.
