Aims.We present results from deep γ-ray observations of the Galactic pulsar wind nebula HESS J1825-137 performed with the HESS array. Methods: . Detailed morphological and spatially resolved spectral studies reveal the very high-energy (VHE) γ-ray aspects of this object with unprecedented precision. Results: . We confirm previous results obtained in a survey of the Galactic Plane in 2004. The γ-ray emission extends asymmetrically to the south and south-west of the energetic pulsar PSR J1826-1334, that is thought to power the pulsar wind nebula. The differential γ-ray spectrum of the whole emission region is measured over more than two orders of magnitude, from 270 GeV to 35 TeV, and shows indications for a deviation from a pure power law. Spectra have also been determined for spatially separated regions of HESS J1825-137. The photon indices from a power-law fit in the different regions show a softening of the spectrum with increasing distance from the pulsar and therefore an energy dependent morphology. Conclusions: . This is the first time that an energy dependent morphology has been detected in the VHE γ-ray regime. The VHE γ-ray emission of HESS J1825-137 is phenomenologically discussed in the scenario where the γ-rays are produced by VHE electrons via Inverse Compton scattering. The high γ-ray luminosity of the source cannot be explained on the basis of constant spin-down power of the pulsar and requires higher injection power in past.