We propose a new method to construct a confidence band for the conditional treatment average treatment effect (CATE), as a function of a continuous covariate in a randomized controlled trial. Our confidence band is bias-aware, taking into account the maximum smoothing bias of the nonparametric estimators used to construct a confidence band. We provide a computationally simple procedure to obtain a bias-aware confidence band whose half-length at each evaluation point is asymptotically shortest uniformly over the domain of the CATE function. The optimality holds over a class of confidence bands that satisfy a set of natural restrictions on the form of the bandwidths used to construct the confidence bands. Using a simulation design mimicking some features of the randomized controlled trial in Bryan et al. (2021), we show that our confidence band performs favorably in terms of the finite sample coverage and the length when compared to the confidence band based on debiased estimators. Additional Monte Carlo simulation results also support this finding. An R package is available for implementing the proposed procedure.