In this paper, we consider two coupled abstract linear evolution equations with one infinite memory acting on the first equation. Our work is motivated by the recent results of [42], where the authors considered the case of two wave equations with one convolution kernel converging exponentially to zero at infinity, and proved the lack of exponential decay. On the other hand, the authors of [42] proved that the solutions decay polynomially at infinity with a decay rate depending on the regularity of the initial data. Under a boundedness condition on the past history data, we prove that the stability of our abstract system holds for convolution kernels having much weaker decay rates than the exponential one. The general and precise decay estimate of solution we obtain depends on the growth of the convolution kernel at infinity, the regularity of the initial data, and the connection between the operators describing the considered equations. We also present various applications to some distributed coupled systems such as wave-wave, Petrovsky-Petrovsky, wave-Petrovsky, and elasticity-elasticity.