The availability of both anatomical connectivity and brain-wide neural activity measurements in C. elegans make the worm a promising system for learning detailed, mechanistic models of an entire nervous system in a data-driven way. However, one faces several challenges when constructing such a model. We often do not have direct experimental access to important modeling details such as single-neuron dynamics and the signs and strengths of the synaptic connectivity. Further, neural activity can only be measured in a subset of neurons, often indirectly via calcium imaging, and significant trial-to-trial variability has been observed. To address these challenges, we introduce a connectome-constrained latent variable model (CC-LVM) of the unobserved voltage dynamics of the entire C. elegans nervous system and the observed calcium signals. We used the framework of variational autoencoders to fit parameters of the mechanistic simulation constituting the generative model of the LVM to calcium imaging observations. A variational approximate posterior distribution over latent voltage traces for all neurons is efficiently inferred using an inference network, and constrained by a prior distribution given by the biophysical simulation of neural dynamics. We applied this model to an experimental whole-brain dataset, and found that connectomic constraints enable our LVM to predict the activity of neurons whose activity were withheld significantly better than models unconstrained by a connectome. We explored models with different degrees of biophysical detail, and found that models with realistic conductance-based synapses provide markedly better predictions than current-based synapses for this system.