This paper presents the private-outsourced-Gaussian process-upper confidence bound (PO-GP-UCB) algorithm, which is the first algorithm for privacy-preserving Bayesian optimization (BO) in the outsourced setting with a provable performance guarantee. We consider the outsourced setting where the entity holding the dataset and the entity performing BO are represented by different parties, and the dataset cannot be released non-privately. For example, a hospital holds a dataset of sensitive medical records and outsources the BO task on this dataset to an industrial AI company. The key idea of our approach is to make the BO performance of our algorithm similar to that of non-private GP-UCB run using the original dataset, which is achieved by using a random projection-based transformation that preserves both privacy and the pairwise distances between inputs. Our main theoretical contribution is to show that a regret bound similar to that of the standard GP-UCB algorithm can be established for our PO-GP-UCB algorithm. We empirically evaluate the performance of our PO-GP-UCB algorithm with synthetic and real-world datasets.