We perform an extensive density matrix renormalization group (DMRG) study of the ground-state phase diagram of the spin-1/2 J_1-J_2 Heisenberg model on the kagome lattice. We focus on the region of the phase diagram around the kagome Heisenberg antiferromagnet, i.e., at J_2=0. We investigate the static spin structure factor, the magnetic correlation lengths, and the spin gaps. Our results are consistent with the absence of magnetic order in a narrow region around J_2approx 0, although strong finite-size effects do not allow us to accurately determine the phase boundaries. This result is in agreement with the presence of an extended spin-liquid region, as it has been proposed recently. Outside the disordered region, we find that for ferromagnetic and antiferromagnetic J_2 the ground state displays signatures of the magnetic order of the sqrt{3}timessqrt{3} and the q=0 type, respectively. Finally, we focus on the structure of the entanglement spectrum (ES) in the q=0 ordered phase. We discuss the importance of the choice of the bipartition on the finite-size structure of the ES.
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