This paper introduces a framework for the automated evaluation of natural language texts. A manually constructed rubric describes how to assess multiple dimensions of interest. To evaluate a text, a large language model (LLM) is prompted with each rubric question and produces a distribution over potential responses. The LLM predictions often fail to agree well with human judges -- indeed, the humans do not fully agree with one another. However, the multiple LLM distributions can be $\textit{combined}$ to $\textit{predict}$ each human judge's annotations on all questions, including a summary question that assesses overall quality or relevance. LLM-Rubric accomplishes this by training a small feed-forward neural network that includes both judge-specific and judge-independent parameters. When evaluating dialogue systems in a human-AI information-seeking task, we find that LLM-Rubric with 9 questions (assessing dimensions such as naturalness, conciseness, and citation quality) predicts human judges' assessment of overall user satisfaction, on a scale of 1--4, with RMS error $< 0.5$, a $2\times$ improvement over the uncalibrated baseline.