We present the first detection of carbon radio recombination line absorption along the line of sight to Cygnus A. The observations were carried out with the Low Frequency Array in the 33-57 MHz range. These low-frequency radio observations provide us with a new line of sight to study the diffuse, neutral gas in our Galaxy. To our knowledge this is the first time that foreground Milky Way recombination line absorption has been observed against a bright extragalactic background source. By stacking 48 carbon α lines in the observed frequency range we detect carbon absorption with a signal-to-noise ratio of about 5. The average carbon absorption has a peak optical depth of 2 × 10-4, a line width of 10 km s-1 and a velocity of +4 km s-1 with respect to the local standard of rest. The associated gas is found to have an electron temperature Te ~ 110K and density ne~0.06 cm-3. These properties imply that the observed carbon α absorption likely arises in the cold neutral medium of the Orion arm of the Milky Way. Hydrogen and helium lines were not detected to a 3σ peak optical depth limit of 1.5 × 10-4 for a 4 km s-1 channel width. Radio recombination lines associated with Cygnus A itself were also searched for, but are not detected. We set a 3σ upper limit of 1.5 × 10-4 for the peak optical depth of these lines for a 4 km s-1 channel width.