The non-thermal nature of the X-ray emission from the shell-type supernova remnants (SNRs) G1.9+0.3 and G330.2+1.0 is an indication of intense particle acceleration in the shock fronts of both objects. This suggests that the SNRs are prime candidates for very-high-energy (VHE; E $>$ 0.1 TeV) {gamma}-ray observations. G1.9+0.3, recently established as the youngest known SNR in the Galaxy, also offers a unique opportunity to study the earliest stages of SNR evolution in the VHE domain. The purpose of this work is to probe the level of VHE {gamma}-ray emission from both SNRs and use this to constrain their physical properties. Observations were conducted with the H.E.S.S. (High Energy Stereoscopic System) Cherenkov telescope array over a more than six-year period spanning 2004-2010. The obtained data have effective livetimes of 67 h for G1.9+0.3 and 16 h for G330.2+1.0. The data are analyzed in the context of the multi-wavelength observations currently available and in the framework of both leptonic and hadronic particle acceleration scenarios. No significant {gamma}-ray signal from G1.9+0.3 or G330.2+1.0 was detected. Upper limits (99% confidence level) to the TeV flux from G1.9+0.3 and G330.2+1.0 for the assumed spectral index {Gamma} = 2.5 were set at 5.6 $times$ 10$^{-13}$ cm$^{-2}$ s$^{-1}$ above 0.26 TeV and 3.2 $times$ 10$^{-12}$ cm$^{-2}$ s$^{-1}$ above 0.38 TeV, respectively. In a one-zone leptonic scenario, these upper limits imply lower limits on the interior magnetic field to B$_{mathrm{G1.9}}$ $gtrsim$ 11 {mu}G for G1.9+0.3 and to B$_{mathrm{G330}}$ $gtrsim$ 8 {mu}G for G330.2+1.0. In a hadronic scenario, the low ambient densities and the large distances to the SNRs result in very low predicted fluxes, for which the H.E.S.S. upper limits are not constraining.
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