The magnetic ground state of the ideal quantum kagome antiferromagnet (QKA) has been a long-standing puzzle, mainly because perturbations to the nearest-neighbor isotropic Heisenberg Hamiltonian can lead to various fundamentally different ground states. Here we investigate a recently synthesized QKA representative YCu$_3$(OH)$_6$Cl$_3$, where perturbations commonly present in real materials, like lattice distortion and intersite ion mixing, are absent. Nevertheless, this compound enters a long-range magnetically ordered state below $T_N=15$ K. Our powder neutron diffraction experiment reveals that its magnetic structure corresponds to a coplanar $120^circ$ state with negative vector spin chirality. The ordered magnetic moments are suppressed to $0.42(2)mu_B$, which is consistent with the previously detected spin dynamics persisting to the lowest experimentally accessible temperatures. This indicates either a coexistence of magnetic order and disorder or the presence of strong quantum fluctuations in the ground state of YCu$_3$(OH)$_6$Cl$_3$. The origin of the magnetic order is sought in terms of Dzyaloshinskii-Moriya magnetic anisotropy and further-neighbor isotropic exchange interactions.