The ternary phase EU3Bi(Sn1-xBix)(4) (similar to 0 < X < similar to 0.15) has been synthesized by solid-state methods at high temperature. The crystal structure of the limiting EU3Bi(Sn3.39Bi0.61(3)) has been determined by single-crystal X-ray analysis to be isopointal with an inverse-Cr5B3-type structure [space group 14/mcm, Z = 4, a = 8.826(1) angstrom, c = 12,564(3) angstrom, and V = 978.6(3) angstrom(3)]. The structure contains slabs of three-bonded Sn/Bi atoms as puckered eight-and four-membered rings interlinked at all vertices, and these are separated by planar layers of individual Eu and Bi atoms. In the normal (stuffed) Cr5B3-type analogue Eu5Sn3Hx, these two units are replaced by a more highly puckered network of Eu cations around isolated Sn atoms and planar layers of isolated Eu atoms and Sn dimers, respectively. Band structures of limiting models of the phase calculated by TB-LMTO-ASA methods show a metallic character and indicate that the mixed Sn/Bi occupancy in the slabs in this structure for x > 0 probably originates with the electronic advantages of the pseudogap that would occur at the electron count of the ideal Zintl phase Eu3Bi(Sn3Bi). The stability of a competing phase reduces this limit to EU3Bi(Sn3.4Bi0.6).