For the weakly interacting one-dimensional multi-particle Anderson model in the continuum space of configurations, we prove the spectral exponential and the strong dynamical localization. The results require the interaction amplitude to be sufficiently small. The general strategy uses the multi-scale analysis bounds. Actually, we show that the multi-scale analysis bounds of the single particle model remain stable when passing to multi-particle systems, provided that the inter-particle interaction is sufficiently small. The common probability distribution of the i.i.d. random external potential in the Anderson model, is only needed to be log-Hölder continuous.