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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.