  In this paper we determine the irreducible components of the Hilbert schemes
H(4,g) of locally Cohen-Macaulay space curves of degree four and arbitrary
arithmetic genus g. We show that these Hilbert schemes are connected, in spite
of having about g^2/24 irreducible components. For g < -2 we exhibit a
component that is disjoint from the component of extremal curves and use this
to give a counterexample to a conjecture of Ait-Amrane and Perrin.
