All 19 literature citations of the lacunary complex PMo11O39(7-) deal with aqueous solutions, conditions under which this anion is kinetically labile. Kinetic stabilization of PMo11O39(7-), a compound of central value in development of the chemistry and applications of defect polymolybdophosphates, is made possible by precipitation of this polyoxoanion from aqueous solutions containing the requisite proportions of the parent Keggin complex alpha-H3PMo12O40, and OH-, by addition of n-Bu4N+. The yield of the product, (Bu4N)4H3PMo11O39 (1), versus the byproducts, principally P2Mo18O62(6-), varies greatly with the initial concentrations of alpha-PMo12O40(3-) and base prior to precipitation. All of the redox couples of 1 in acetonitrile are completely irreversible, unlike those of the parent complex, alpha-H3PMo12O40, and other polyoxometalates whose structures are comprised of MO6 octahedra with one terminal oxo group. In contrast to PMo11O39(7-) in aqueous solution, 1 in acetonitrile solution can be (a) highly purified by recrystallization with no generation of other polymolybdophosphates, (b) reversibly protonated to H4PMo11O39(3-), which slowly decomposes to PMo12O40(3-), P2Mo18O62(6-), and P2Mo5O23(6-), and (c) reversibly deprotonated to H2PMo11O39(5-) by 1 equiv of OH- and readily converted to alpha-A-PMo9O31(OH)3(6-) and PMo12O40(3-) by > 1 equiv of OH-. 1 can be metalated with several divalent d(n), n not-equal 0 transition-metal (TM) ions to form the corresponding TM-substituted polymolybophosphates, PMo11(TM)O39(5-). Unlike Co(II), Mn(II), or Cu(II), metalation with Zn(II) induces the decomposition of 1. Effective metalations of 1 can be carried out by a homogeneous one-phase procedure (acetonitrile solvent and TM triflate salt) or by a heterogeneous two-phase procedure (treatment of 3:2 acetonitrile-toluene solution of 1 with an aqueous solution of the TM salt). The rates of exchange of terminal axial ligands, L, in PM11TM-L)O39n- (M = Mo, W), in the aprotic medium, acetonitrile, which depend on the TM, the axial ligand, and on the polyoxometalate ligand environment, are discussed.