A measurement of the underlying activity in scattering processes with transverse momentum scale in the GeV region is performed in proton-proton collisions at sqrt(s) = 0.9 TeV, using data collected by the CMS experiment at the LHC. Charged hadron production is studied with reference to the direction of a leading object, either a charged particle or a set of charged particles forming a jet. Predictions of several QCD-inspired models as implemented in PYTHIA are compared, after full detector simulation, to the data. The models generally predict too little production of charged hadrons with pseudorapidity eta < 2, p_T > 0.5 GeV/c, and azimuthal direction transverse to that of the leading object.