Using the Karl G. Jansky Very Large Array, we have detected absorption lines due to carbon monoxide, CO(J = 0 → 1), and the cyano radical, CN(N = 0 → 1), associated with radio galaxy B2 0902+34 at redshift z = 3.4. The detection of millimeter-band absorption observed 1.5 Gyr after the Big Bang facilitates studying molecular clouds down to gas masses inaccessible to emission-line observations. The CO absorption in B2 0902+34 has a peak optical depth of τ ≥ 8.6% and consists of two components, one of which has the same redshift as previously detected 21 cm absorption of neutral hydrogen (H i) gas. Each CO component traces an integrated H2 column density of
N
H
2
≳ 3 × 1020 cm−2. CN absorption is detected for both CO components, as well as for a blueshifted component not detected in CO, with CO/CN line ratios ranging from ≲0.4 to 2.4. We discuss the scenario that the absorption components originate from collections of small and dense molecular clouds that are embedded in a region with more diffuse gas and high turbulence, possibly within the influence of the central active galactic nucleus or a starburst region. The degree of reddening in B2 0902+34, with rest-frame color B − K ∼ 4.2, is lower than the very red colors (B − K > 6) found among other known redshifted CO absorption systems at z < 1. Nevertheless, when including the many nondetections from the literature, a potential correlation between the absorption-line strength and B − K color is evident, giving weight to the argument that the red colors of CO absorbers are due to a high dust content.