The role of antiferromagnetic spin correlations in high-temperature superconductors remains a matter of debate. We present inelastic neutron scattering evidence that gapless spin fluctuations coexist with superconductivity in La1.905Ba0.095CuO4. Furthermore, we observe that both the lowenergy magnetic spectral weight and the spin incommensurability are enhanced with the onset of superconducting correlations. We propose that the coexistence occurs through intertwining of spatial modulations of the pair wave function and the antiferromagnetic correlations. This proposal is also directly relevant to sufficiently underdoped La2−xSrxCuO4 and YBa2Cu3O6+x.It is commonly accepted that cuprate superconductors have a spatially-uniform d-wave pair wave function [1]. It has also become a paradigm that antiferromagnetic spin fluctuations are gapped in the superconducting state, with a pile up of excitations in the magnetic "resonance" peak above the gap [2][3][4][5][6]. A number of neutron scattering studies of underdoped La 2−x Sr x CuO 4 have found evidence for incommensurate spin fluctuations that remain gapless at temperatures far below the superconducting transition temperature, T c [7][8][9][10]. Theoretical analyses have tended to view such spin-density-wave correlations as soft fluctuations of an order that competes with spatially-uniform superconductivity [11] and that may be locally pinned by disorder [12]. As a consequence, researchers have crafted interpretations of the low-energy spin fluctuations that maintain consistency with the spingap paradigm [8,10].In an alternative approach, the superconductivity and antiferromagnetism are both treated as spatially modulated and intimately intertwined [13]. Such a state, which variational calculations indicate to be energetically competitive with uniform superconductivity [14], has been invoked [15,16] to explain the depression of superconducting order in certain stripe-ordered cuprates [17,18]. While the poorly-superconducting phase is fascinating on its own, it leaves open the question of whether a modulated pair wave function might be relevant to the case of a good bulk superconductor.In this paper, we present neutron scattering measurements of the low-energy spin fluctuations in La 1.905 Ba 0.095 CuO 4 , a bulk superconductor with T c = 32 K. Rather than developing a spin gap on cooling below T c , the lowest-energy excitations are actually enhanced. By putting the measurements on an absolute scale, we show that the strength of the spin response is comparable to that of spin waves in antiferromagnetic La 2 CuO 4 . To generate this large a response, we conclude that all parts of the sample must contribute to the signal, ruling out macroscopic phase separation. A previous optical conductivity study has shown that the superfluid density of this sample is consistent with the trend established for bulk superconductivity in all cuprate families in the form of Homes' law [19]. It thus appears that there must be local coexistence of the spin fluctuations and superconductivit...