Pre-trained large language models (LLMs) reflect the inherent social biases of their training corpus. Many methods have been proposed to mitigate this issue, but they often fail to debias or they sacrifice model accuracy. We use *conceptors*–a soft projection method–to identify and remove the bias subspace in LLMs such as BERT and GPT. We propose two methods of applying conceptors (1) bias subspace projection by post-processing by the conceptor NOT operation; and (2) a new architecture, conceptor-intervened BERT (CI-BERT), which explicitly incorporates the conceptor projection into all layers during training. We find that conceptor post-processing achieves state-of-the-art (SoTA) debiasing results while maintaining LLMs’ performance on the GLUE benchmark. Further, it is robust in various scenarios and can mitigate intersectional bias efficiently by its AND operation on the existing bias subspaces. Although CI-BERT’s training takes all layers’ bias into account and can beat its post-processing counterpart in bias mitigation, CI-BERT reduces the language model accuracy. We also show the importance of carefully constructing the bias subspace. The best results are obtained by removing outliers from the list of biased words, combining them (via the OR operation), and computing their embeddings using the sentences from a cleaner corpus.