This research presents a generative Artificial Intelligence (AI) and design framework that integrates machine learning (ML) and optimisation methodologies to discover new concrete mixture designs. Unlike traditional ML models that predict based on existing data, this framework innovatively generates new concrete mix designs that meet specific requirements such as strength, cost-efficiency, and reduced embodied CO2. To propose a powerful and reliable generative AI model, several advanced ML algorithms were considered, e.g., CatBoost, XGBoost, and LGBM. These models were trained on a unique dataset consisting of 4,936 data points collected from five different batching plants and have not been published yet. Bayesian Optimisation was employed to fine-tune model hyperparameters, resulting in the most effective models attaining R2 values of 0.94 and 0.89 for raw and grouped data, respectively. To verify the trained generative AI model, a case study was conducted, in which the model was requested to provide designs of a mix with pre-determined strength and optimised cost and embodied CO2. The mix designs generated by the framework were successfully validated through experimental tests, corroborating the predictive outcomes. The research culminated in the development of a web application, a tool crafted to streamline the concrete mixture design and optimisation process. This generative AI design framework can be applied to many other aspects of material design and engineering problems.