Polymers from norbornenes are of interest for applications in opto- and microelectronic (low dielectric materials, photoresists, OLEDs). Norbornenes with ester motifs are among the most readily available norbornene derivatives. However, little is known about dielectric properties and the gas-transport of polynorbornenes from such monomers. Herein, we synthesized a new metathesis polymer from exo-5-norbornenecarboxylic acid and 1,1′-bi-2-naphthol. The designed monomer was obtained via a two-step procedure in a good yield. This norbornene derivative with a rigid and a bulky binaphthyl group was successfully polymerized over the 1st generation Grubbs catalyst, affording high-molecular-weight products (Mw ≤ 1.5·106) in yields of 94–98%. The polymer is amorphous and glassy (Tg = 161 °C), and it shows good thermal stability. Unlike most, polyNBi is a classic low-permeable glassy polymer. The selectivity of polyNBi was higher than that of polyNB. Being less permeable than polyNB, polyNBi unexpectedly showed a lower value of dielectric permittivity (2.7 for polyNBi vs. 5.0 for polyNB). Therefore, the molecular design of polynorbornenes has great potential to obtain polymers with desired properties in a wide range of required characteristics. Further tuning of the gas separation efficiency can be achieved by attaching an appropriate substituent to the ester and aryl group.