Large language models (LLMs) have demonstrated remarkable advancements in language understanding and generation. Building on the success of text-based LLMs, recent research has adapted these models to use speech embeddings for prompting, resulting in Speech-LLM models that exhibit strong performance in automatic speech recognition (ASR) and automatic speech translation (AST). In this work, we propose a novel approach to leverage ASR transcripts as prompts for AST in a Speech-LLM built on an encoder-decoder text LLM. The Speech-LLM model consists of a speech encoder and an encoder-decoder structure Megatron-T5. By first decoding speech to generate ASR transcripts and subsequently using these transcripts along with encoded speech for prompting, we guide the speech translation in a two-step process like chain-of-thought (CoT) prompting. Low-rank adaptation (LoRA) is used for the T5 LLM for model adaptation and shows superior performance to full model fine-tuning. Experimental results show that the proposed CoT prompting significantly improves AST performance, achieving an average increase of 2.4 BLEU points across 6 En->X or X->En AST tasks compared to speech prompting alone. Additionally, compared to a related CoT prediction method that predicts a concatenated sequence of ASR and AST transcripts, our method performs better by an average of 2 BLEU points.