Fine-tuning large language models (LLMs) with low-rank adaptation (LoRA) is a cost-effective way to incorporate information from a specific dataset. However, when a problem requires incorporating information from multiple datasets - as in few shot learning - generalization across datasets can be limited, driving up training costs. As a consequence, other approaches such as in-context learning are typically used in this setting. To address this challenge, we introduce an efficient method for adapting the weights of LLMs to multiple distributions, Amortized Bayesian Meta-Learning for LoRA (ABMLL). This method builds on amortized Bayesian meta-learning for smaller models, adapting this approach to LLMs by reframing where local and global variables are defined in LoRA and using a new hyperparameter to balance reconstruction accuracy and the fidelity of task-specific parameters to the global ones. ABMLL supports effective generalization across datasets and scales to large models such as Llama3-8B and Qwen2-7B, outperforming existing methods on the CrossFit and Unified-QA datasets in terms of both accuracy and expected calibration error. We show that meta-learning can also be combined with in-context learning, resulting in further improvements in both these datasets and legal and chemistry applications.