The gamma-ray pulsar PSR B1706-44 and the adjacent supernova remnant (SNR) candidate G 343.1-2.3 were observed by H. E. S. S. during a dedicated observation campaign in 2007. As a result of this observation campaign, a new source of very-high-energy (VHE; E > 100 GeV) gamma-ray emission, H.E.S.S. J1708-443, was detected with a statistical significance of 7 sigma, although no significant point-like emission was detected at the position of the energetic pulsar itself. In this paper, the morphological and spectral analyses of the newly-discovered TeV source are presented. The centroid of H. E. S. S. J1708-443 is considerably offset from the pulsar and located near the apparent center of the SNR, at alpha(J2000) = 17(h)08(m)11(s) +/- 17(s) and delta(J2000) = -44 degrees 20' +/- 4'. The source is found to be significantly more extended than the H. E. S. S. point spread function (similar to 0.1 degrees), with an intrinsic Gaussian width of 0.29 degrees +/- 0.04 degrees. Its integral flux between 1 and 10 TeV is similar to 3.8 x 10(-1)2 ph cm(-2) s(-1), equivalent to 17% of the Crab Nebula flux in the same energy range. The measured energy spectrum is well-fit by a power law with a relatively hard photon index Gamma = 2.0 +/- 0.1(stat) +/-0.2(sys). Additional multi-wavelength data, including 330 MHz VLA observations, were used to investigate the VHE gamma-ray source's possible associations with the pulsar wind nebula of PSR B1706-44 and/or with the complex radio structure of the partial shell-type SNR G 343.1-2.3.