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Recent studies have found interference cancellation (IC) schemes employing blind source separation (BSS) methods to be promising competitors to conventional IC schemes in DS-CDMA reception, since they are able to mitigate both internal multiaccess interference and external interference. In this paper we introduce a new successive BSS based IC scheme for DS-CDMA uplink. The new scheme combines the benefits of existing BSS based schemes and conventional successive interference cancellation (SIC). Numerical experiments are given to illustrate a superiority of the successive BSS scheme compared to existing BSS schemes and conventional successive and parallel interference cancellation (PIC) schemes. A system with and without additional external interference, like adjacent channel interference or jamming, are considered.