The rise of big data has amplified the need for efficient, user-friendly automated machine learning (AutoML) tools. However, the intricacy of understanding domain-specific data and defining prediction tasks necessitates human intervention making the process time-consuming while preventing full automation. Instead, envision an intelligent agent capable of assisting users in conducting AutoML tasks through intuitive, natural conversations without requiring in-depth knowledge of the underlying machine learning (ML) processes. This agent's key challenge is to accurately comprehend the user's prediction goals and, consequently, formulate precise ML tasks, adjust data sets and model parameters accordingly, and articulate results effectively. In this paper, we take a pioneering step towards this ambitious goal by introducing a ChatGPT-based conversational data-science framework to act as a "personal data scientist". Precisely, we utilize Large Language Models (ChatGPT) to build a natural interface between the users and the ML models (Scikit-Learn), which in turn, allows us to approach this ambitious problem with a realistic solution. Our model pivots around four dialogue states: Data Visualization, Task Formulation, Prediction Engineering, and Result Summary and Recommendation. Each state marks a unique conversation phase, impacting the overall user-system interaction. Multiple LLM instances, serving as "micro-agents", ensure a cohesive conversation flow, granting us granular control over the conversation's progression. In summary, we developed an end-to-end system that not only proves the viability of the novel concept of conversational data science but also underscores the potency of LLMs in solving complex tasks. Interestingly, its development spotlighted several critical weaknesses in the current LLMs (ChatGPT) and highlighted substantial opportunities for improvement.