  The kinematics of two-neutron emission following the $\beta$-decay of
$^{11}$Li was investigated for the first time by detecting the two neutrons in
coincidence and by measuring their angle and energy. An array of
liquid-scintillator neutron detectors was used to reject cosmic-ray and
$\gamma$-ray backgrounds by pulse-shape discrimination. Cross-talk events in
which two detectors are fired by a single neutron were rejected using a filter
tested on the $\beta$-1n emitter $^9$Li. A large cross-talk rejection rate is
obtained ($> 95 \%$) over most of the energy range of interest. Application to
$^{11}$Li data leads to a significant number of events interpreted as
$\beta$-2n decay. A discrete neutron line at $\approx$ 2 MeV indicates
sequential two-neutron emission, possibly from the unbound state at 10.6 MeV
excitation energy in $^{11}$Be.
