  We present gapless, high-resolution absorption and dispersion spectra
obtained with quantum cascade laser frequency combs covering 55 cm$^{-1}$.
Using phase-sensitive dual comb design, the comb lines are gradually swept over
10 GHz, corresponding to the free spectral range of the laser devices, by
applying a current modulation. We show that with interleaving the spectral
point spacing is reduced by more than four orders of magnitude over the full
spectral span of the frequency comb. The potential of this technique for
high-precision gas sensing is illustrated by measuring the low pressure (107
hPa) absorption and dispersion spectra of methane spanning the range of 1170
cm$^{-1}$ - 1225 cm$^{-1}$ with a resolution of 0.001 cm$^{-1}$.
