Bayesian Optimization (BO) has been widely used to efficiently optimize expensive black-box functions with limited evaluations. In this paper, we investigate the use of BO for prompt engineering to enhance text classification with Large Language Models (LLMs). We employ an LLM-powered Gaussian Process (GP) as the surrogate model to estimate the performance of different prompt candidates. These candidates are generated by an LLM through the expansion of a set of seed prompts and are subsequently evaluated using an Upper Confidence Bound (UCB) acquisition function in conjunction with the GP posterior. The optimization process iteratively refines the prompts based on a subset of the data, aiming to improve classification accuracy while reducing the number of API calls by leveraging the prediction uncertainty of the LLM-based GP. The proposed BO-LLM algorithm is evaluated on two datasets, and its advantages are discussed in detail in this paper.