  We study the quantum phase diagram of the spin-$1/2$ Heisenberg model on the
kagom\'e lattice with first-, second-, and third-neighbor interactions $J_1$,
$J_2$, and $J_3$ by means of density matrix renormalization group. For small
$J_2$ and $J_3$, this model sustains a time-reversal invariant quantum spin
liquid phase. With increasing $J_2$ and $J_3$, we find in addition a $q=(0,0)$
N\'{e}el phase, a chiral spin liquid phase, a valence-bond crystal phase, and a
complex non-coplanar magnetically ordered state with spins forming the vertices
of a cuboctahedron known as a cuboc1 phase. Both the chiral spin liquid and
cuboc1 phase break time reversal symmetry in the sense of spontaneous scalar
spin chirality. We show that the chiralities in the chiral spin liquid and
cuboc1 are distinct, and that these two states are separated by a strong first
order phase transition. The transitions from the chiral spin liquid to both the
$q=(0,0)$ phase and to time-reversal symmetric spin liquid, however, are
consistent with continuous quantum phase transitions.
