  We use the Low Frequency Array to perform a systematic high spectral
resolution investigation of the low-frequency 33-78 MHz spectrum along the line
of sight to Cassiopeia A. We complement this with a 304-386 MHz Westerbork
Synthesis Radio telescope observation. In this first paper we focus on the
carbon radio recombination lines.
  We detect Cn$\alpha$ lines at -47 and -38 km s$^{-1}$ in absorption for
quantum numbers n=438-584 and in emission for n=257-278 with high signal to
noise. These lines are associated with cold clouds in the Perseus spiral arm
component. Hn$\alpha$ lines are detected in emission for n=257-278. In
addition, we also detect Cn$\alpha$ lines at 0 km s$^{-1}$ associated with the
Orion arm.
  We analyze the optical depth of these transitions and their line width. Our
models show that the carbon line components in the Perseus arm are best fit
with an electron temperature 85 K and an electron density 0.04 cm$^{-3}$ and
can be constrained to within 15\%. The electron pressure is constrained to
within 20\%. We argue that much of these carbon radio recombination lines arise
in the CO-dark surface layers of molecular clouds where most of the carbon is
ionized but hydrogen has made the transition from atomic to molecular. The
hydrogen lines are clearly associated with the carbon line emitting clouds, but
the low-frequency upperlimits indicate that they likely do not trace the same
gas. Combining the hydrogen and carbon results we arrive at a firm lower limit
to the cosmic ray ionization rate of 2.5$\times$10$^{-18}$ s$^{-1}$, but the
actual value is likely much larger.
