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In this paper, we provide a transform from an $\varepsilon$-BIC mechanism into an exactly BIC mechanism without any loss of social welfare and with additive and negligible revenue loss. This is the first $\varepsilon$-BIC to BIC transformation that preserves welfare and provides negligible revenue loss. The revenue loss bound is tight given the requirement to maintain social welfare. Previous $\varepsilon$-BIC to BIC transformations preserve social welfare but have no revenue guarantee~\citep{BeiHuang11}, or suffer welfare loss while incurring a revenue loss with both a multiplicative and an additive term, e.g.,~\citet{DasWeinberg12, Rubinstein18, Cai19}. The revenue loss achieved by our transformation is incomparable to these earlier approaches and can be significantly less. \newnew{Our approach is different from the previous replica-surrogate matching methods and we directly make use of a directed and weighted type graph (induced by the types' regret), one for each agent. The transformation runs a \emph{fractional rotation step} and a \emph{payment reducing step} iteratively to make the mechanism Bayesian incentive compatible.} We also analyze $\varepsilon$-expected ex-post IC ($\varepsilon$-EEIC) mechanisms~\citep{DuettingFJLLP12}. We provide a welfare-preserving transformation in this setting with the same revenue loss guarantee for uniform type distributions and give an impossibility result for non-uniform distributions. We apply the transform to linear-programming based and machine-learning based methods of automated mechanism design.