  We present a Chandra X-ray observation of the very high energy $\gamma$-ray
source HESS$ $J1640-465. We identify a point source surrounded by a diffuse
emission that fills the extended object previously detected by XMM Newton at
the centroid of the HESS source, within the shell of the radio supernova
remnant (SNR) G338.3-0.0. The morphology of the diffuse emission strongly
resembles that of a pulsar wind nebula (PWN) and extends asymmetrically to the
South-West of a point-source presented as a potential pulsar. The spectrum of
the putative pulsar and compact nebula are well-characterized by an absorbed
power-law model which, for a reasonable $N_{\rm H}$ value of $14\times 10^{22}
\rm cm^{-2}$, exhibit an index of 1.1 and 2.5 respectively, typical of
Vela-like PWNe. We demonstrate that, given the H$ $I absorption features
observed along the line of sight, the SNR and the H$ $II surrounding region are
probably connected and lie between 8 kpc and 13 kpc. The resulting age of the
system is between 10 and 30 kyr. For a 10 kpc distance (also consistent with
the X-ray absorption) the 2-10 keV X-ray luminosities of the putative pulsar
and nebula are $L_{\rm PSR} \sim 1.3 \times 10^{33} d_{10 \rm kpc}^{2} \rm
erg.s^{-1}$ and $L_{\rm PWN} \sim 3.9 \times 10^{33} d_{10}^{2} \rm erg.s^{-1}$
($d_{10} = d / 10{\rm kpc}$). Both the flux ratio of $L_{\rm PWN}/L_{\rm PSR}
\sim 3.4$ and the total luminosity of this system predict a pulsar spin-down
power around $\dot{E} \sim 4 \times 10^{36} \rm erg s^{-1}$. We finally
consider several reasons for the asymmetries observed in the PWN morphology and
discuss the potential association with the HESS source in term of a
time-dependent one-zone leptonic model.
