Modern hardware design starts with specifications provided in natural language. These are then translated by hardware engineers into appropriate Hardware Description Languages (HDLs) such as Verilog before synthesizing circuit elements. Automating this translation could reduce sources of human error from the engineering process. But, it is only recently that artificial intelligence (AI) has demonstrated capabilities for machine-based end-to-end design translations. Commercially-available instruction-tuned Large Language Models (LLMs) such as OpenAI's ChatGPT and Google's Bard claim to be able to produce code in a variety of programming languages; but studies examining them for hardware are still lacking. In this work, we thus explore the challenges faced and opportunities presented when leveraging these recent advances in LLMs for hardware design. Using a suite of 8 representative benchmarks, we examined the capabilities and limitations of the state of the art conversational LLMs when producing Verilog for functional and verification purposes. Given that the LLMs performed best when used interactively, we then performed a longer fully conversational case study where a hardware engineer co-designed a novel 8-bit accumulator-based microprocessor architecture. We sent the benchmarks and processor to tapeout in a Skywater 130nm shuttle, meaning that these 'Chip-Chats' resulted in what we believe to be the world's first wholly-AI-written HDL for tapeout.