  We provide an experimental demonstration that the focusing of ionic currents
in a micron size hole connecting two chambers can produce local temperature
increases of up to $100^\circ$ C with gradients as large as $1^\circ$ K$\mu
m^{-1}$. We find a good agreement between the measured temperature profiles and
a finite elements-based numerical calculation. We show how the thermal
gradients can be used to measure the full melting profile of DNA duplexes
within a region of 40 $\mu$m. The possibility to produce even larger gradients
using sub-micron pores is discussed.
