Two-dimensional (2D) $A$-type antiferromagnetic van der Waals (vdW) materials with either intralayer ferromagnetic and/or intralayer antiferromagnetic coupling have sparked great interest due to their potential applications in spintronics. By first-principles design we predict that a heterostructure formed by sandwiching the 2D $A$-type antiferromagnet, $2\mathrm{H}\text{\ensuremath{-}}\mathrm{V}{\mathrm{Se}}_{2}$, between the 2D out-of-plane ferroelectric, ${\mathrm{Sc}}_{2}{\mathrm{CO}}_{2}$, presents a semimetal band structure with a half-metallic conduction band. The mechanism giving rise to this peculiar structure cooperatively originates from the strong built-in electric field of the double-layer ferroelectric ${\mathrm{Sc}}_{2}{\mathrm{CO}}_{2}$, and from the charge transfer selectively occurring only at one interface. Based on the so-designed ${\mathrm{Sc}}_{2}{\mathrm{CO}}_{2}\text{/}\mathrm{V}{\mathrm{Se}}_{2}$ multiferroic heterostructure, two concepts for nonvolatile memory devices are proposed, in which two states (``1'' and ``0'') are realized by switching the polarization direction of the ferroelectric layers. These results demonstrate that the combination of 2D ferroelectrics and $A$-type antiferromagnetic vdW materials provides not only a fascinating way to achieve half-metallicity in 2D materials, but also a route to the design of new types of nonvolatile ferroelectric memory devices.