The production of mesons containing strange quarks (K⁰ <inf>S</inf>, ϕ) and both singly and doubly strange baryons (⋀, ⊼, and Ξ⁻ +Ξ⁺) are measured at mid-rapidity in pp collisions at √s = 0.9 TeV with the ALICE experiment at the LHC. The results are obtained from the analysis of about 250 k minimum bias events recorded in 2009. Measurements of yields (dN/dy) and transverse momentum spectra at mid-rapidity for inelastic pp collisions are presented. For mesons, we report yields (〈dN/dy〉) of 0.184 ± 0.002(stat.) ± 0.006(syst.) for K⁰ <inf>S</inf> and 0.021 ± 0.004(stat.) ± 0.003(syst.) for ϕ. For baryons, we find 〈dN/dy〉 = 0.048 ± 0.001(stat.) ± 0.004(syst.) for ⋀, 0.047 ± 0.002(stat.) ± 0.005(syst.) for ⊼ and 0.0101 ± 0.0020(stat.) ± 0.0009(syst.) for Ξ⁻ +Ξ⁺. The results are also compared with predictions for identified particle spectra from QCD-inspired models and provide a baseline for comparisons with both future pp measurements at higher energies and heavy-ion collisions.