The recent discovery of magnetism in atomically thin layers of van der Waals (vdW) crystals has created new opportunities for exploring magnetic phenomena in the two-dimensional (2D) limit. In most 2D magnets studied to date, the c-axis is an easy axis, so that at zero applied field the polarization of each layer is perpendicular to the plane. Here, we demonstrate that atomically thin CrCl<sub>3</sub> is a layered antiferromagnetic insulator with an easy-plane normal to the c-axis, that is, the polarization is in the plane of each layer and has no preferred direction within it. Ligand-field photoluminescence at 870 nm is observed down to the monolayer limit, demonstrating its insulating properties. We investigate the in-plane magnetic order using tunneling magnetoresistance in graphene/CrCl<sub>3</sub>/graphene tunnel junctions, establishing that the interlayer coupling is antiferromagnetic down to the bilayer. From the temperature dependence of the magnetoresistance, we obtain an effective magnetic phase diagram for the bilayer. Our result shows that CrCl<sub>3</sub> should be useful for studying the physics of 2D phase transitions and for making new kinds of vdW spintronic devices.