  The $4f$-electron system YbAl$_3$C$_3$ with a non-magnetic spin-dimer ground
state has been studied by neutron diffraction in an applied magnetic field. A
long-range magnetic order involving both ferromagnetic and antiferromagnetic
components has been revealed above the critical field H$_C\sim $ 6T at
temperature T=0.05K. The magnetic structure indicates that the geometrical
frustration of the prototype hexagonal lattice is not fully relieved in the
low-temperature orthorhombic phase. The suppression of magnetic ordering by the
remanent frustration is the key factor stabilizing the non-magnetic singlet
ground state in zero field. Temperature dependent measurements in the applied
field H=12T revealed that the long-range ordering persists up to temperatures
significantly higher than the spin gap indicating that this phase is not
directly related to the singlet-triplet excitation. Combining our neutron
diffraction results with the previously published phase diagram, we support the
existence of an intermediate disordered phase as the first excitation from the
non-magnetic singlet ground state. Based on our results, we propose
YbAl$_3$C$_3$ as a new material for studying the quantum phase transitions of
heavy-fermion metals under the influence of geometrical frustration.
