The infinite-size version of the density-matrix renormalization-group approach in real space is applied to the Kondo lattice model in one dimension, providing a more accurate determination of the zero-temperature magnetic properties for electron densities n less than or equal to 0.5. The paramagnetic state of the free model evolves into a state with ferromagnetic correlations, by increasing the strength of the Kondo coupling, in agreement with previous numerical results. The change in the magnetic properties is produced by the competition between the RKKY mechanism, which favours the paramagnetic state in the weak-coupling region, and the Kondo screening, which leads to the formation of tightly bound singlets in the strong-coupling region, enforcing ferromagnetic correlations in the system.