Short Answer:

The fruiting body or fruit body in fungi is called the sporocarp.

Details:

When most people see a sporocarp they call this a mushroom, but this fleshy fruiting body is only the visible part of the living organism that is popular for eating. The fruiting body only develops as part of the asexual phase of the fungal life cycle for spore production. To get more specific about the body parts of a mushroom, the fruiting body of the most common mushrooms have a stem (stipe), a cap (pileus), and gills (lamellae).

Indeed, since we are getting technical about it, there are two kinds of sporocarp that most people recognize.

The typical toadstool mushroom is a basidiomycete and the sporocarp is a basidiocarp or basidiome.

Both the popular morel mushroom and the truffle are of the type known as an ascomycete and the fruiting body is an ascocarp.

Short answer:

The part of a fungus that produces spores is the fruiting body.

Long Answer:

A sporangium is a structure in which spores are formed. Plants, fungi, and some other organisms form sporangia.

When most people see a sporocarp they call this a mushroom or toadstool. This fleshy fruiting body is only the visible part of the living organism that is popular for eating. The fruiting body only develops as part of the sexual phase of the fungal life cycle for spore production.

In fungi, the sporocarp is the fruiting body (or fruit body). It is the large structure that contains the smaller contains spore-producing structures, such as basidia or asci. A basidium usually bears four (sometimes eight or more) sexual spores. An ascus, in ascomycete fungi have typically eight ascospores, but some species have other numbers.