Grid turbulence experiments have been carried out in a stably stratified fluid at moderately large Reynolds numbers (160 based on the Taylor microscale). A scanning particle image velocimetry technique is used to provide time-resolved velocity fields in a relatively large volume. For late times, in the low-Froude-number regime, the flow consists of quasi-horizontal motion in a sea of weak internal gravity waves. In this regime the dynamics of the flow is found to be independent of the ambient stratification. Fundamental differences with two-dimensional turbulence, due to the strong vertical shearing of horizontal velocity, are observed. In this regime, a self-similar scaling law for the energy decay and the length-scale evolution are observed. This behaviour reflects a process of adjustment of the eddy aspect ratio based on a balance between the horizontal advective motion which tends to vertically decorrelate the flow and the dissipation due to the strong vertical shear. The characteristic vertical size of the eddies grows according to a diffusion law and is found to be independent of the turbulence generation. The organization of the flow into horizontal layers of eddies separated by intense shear leads to a strong anisotropy of the dissipation: this has been checked by direct measurement of the different tensorial components of the viscous dissipation.