Let B be a closed dissipative operator in a Hilbert space H with an arbitrary domain of definition. We will give a description of all closed (and, in particular, closed maximal) dissipative extensions\(\tilde B\) of B in terms of extensions\(\tilde W\) of a nonexpanding operator W associated with B. We construct a family {Bz} of maximal closed dissipative extensions of B, where z is a complex number in the lower half-plane. We present an example which illustrates the above concepts.