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This paper presents the first observations of molecular gas associated with the Wolf-Rayet ring nebula NGC 3199 around the WR star WR 18. This includes first observations of the molecules HCN, HCO⁺, CN, and HNC seen in any Wolf-Rayet ring nebula. Our observations immediately suggest the presence of high-density molecular gas (>10⁴ cm^–3) in the nebula with significant amounts of associated molecular gas, which is in the form of clumpy ejecta and/or interstellar material. Molecular CO gas was mapped across the optically bright portion of the nebula and out into the diffuse ionized component using the ¹²CO J=1⟶0 line. CO gas is not seen within the optically bright rim of NGC 3199 but adjacent to it. The optical emission rim therefore appears to mark regions of photodissociation. Velocity components in the CO data are consistent with those seen in high-resolution optical spectra of the Hα line but extend beyond the visible emission. A prior suggestion of the formation of the nebula via a bow shock appears unlikely since Hipparcos measurements show the proper motion of WR 18 is almost at right angles to the direction required for the bow shock model. Instead, line splitting toward the north of the nebula suggests that a possible blowout of the Wolf-Rayet wind through surrounding ejecta may be responsible for some of the velocity features observed. Preliminary estimates of molecular abundances in the nebula seen toward the central star are significantly higher than for the interstellar medium and are similar to those in planetary nebulae, although CN is distinctly underabundant in comparison to the very high values found in many planetary nebulae. The abundances found are consistent with the idea that at least a portion of the molecular material is associated with ejecta from the central star.