  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.
