Federated learning (FL) enables the collaborative training of large-scale language models (LLMs) across edge devices while keeping user data on-device. However, FL still exposes sensitive information through client-provided gradients. Differentially private stochastic gradient descent (DP-SGD) mitigates this risk by clipping each client's contribution to a threshold $C$ and adding noise proportional to $C$. Existing adaptive clipping techniques dynamically adjust $C$ but demand tedious hyperparameter tuning, which can erode the privacy budget. In this paper, we introduce DP-LAC, a method that first estimates an initial clipping threshold within an order of magnitude of the optimum using private histogram estimation, and then adapts this threshold during training without consuming additional privacy budget or introducing new hyperparameters. Empirical results show that DP-LAC outperforms both state-of-the-art adaptive clipping methods and vanilla DP-SGD, achieving an average accuracy gain of $6.6\%$.