This dissertation concerns the hydrography of hydrothermal plumes and the problem of measuring the flux of heat from submarine volcanoes to the deep sea. It presents the results of an experiment designed to measure the heat flux from the Main Endeavour hydrothermal vent field (MEF) on the Endeavour segment of the Juan de Fuca ridge in the northeast Pacific. Primary data are from an underwater vehicle called Autonomous Benthic Explorer (ABE), a lowered CTD, and 2 current meter moorings. Context for the experiment is established in Chapter~\ref{ch:power} through motivating questions, definition of terminology, re-examination of plume theory, and review of past heat flux measurements at the MEF. Chapter~\ref{ch:flowmow} presents the experimental setting, methodology, and an analysis of currents and hydrography near the MEF. A key observation is that axial currents change from rectified, oscillatory flow within the axial valley to rotary flow above the ridge. Consequently, the problem of estimating heat flux in addressed separately in the 2 environments: within the axial valley (Chapter~\ref{ch:lower}) and above the ridge (Chapter~\ref{ch:upper}). In both, a ``puff'' model of advection and diffusion simulates plume distributions generated by a point source venting into combinations of oscillatory and mean flow. Within the axial valley, the north end of the MEF is warmer on average than the south end, despite variability induced by tidal oscillations with $\sim$5~cm/s amplitude. The temperature difference and a northward mean flow of 2--5~cm/s yield a mean horizontal flux of $\sim$65~MW. Hydrography varies on unexpectedly short scales of 20~m and 10~min. The puff model indicates the standard deviation of the mean horizontal flux is high ($\sim$100~MW). Vertical velocity and hydrography in rising plumes surveyed $\sim$100~m above the MEF lead to a high-precision estimate of vertical heat flux: 550$\pm$100~MW \cite{stahr+03}. Combination of this flux, past source flux measurements, and the horizontal flux estimate implies that heat flux partitioning between focused and diffuse sources is about 1:1, and that $\sim$4/5 of diffuse heat flux is entrained. Above the ridge, equilibrating plumes and currents generate hydrography that varies on scales as short as 10--50\,m and 1\,hr. The mean observed net horizontal flux is 315$\pm$651\,MW. The puff model explains how the instantaneous net horizontal flux can differ from the underlying source flux by as much as an order of magnitude. Isohaline temperature anomalies are used to define plume distributions and to estimate a plume heat flux that is related to the source heat flux through a factor that accounts for entrainment and source properties. Given the ambient conditions and source salinity anomalies at the MEF, the above heat fluxes may need to be multiplied by a factor of up to 1.4.