  The thin string limit of Cosmic Strings is investigated using a description
in terms of Colombeau's theory of nonlinear generalised functions. It is shown
that in this description the energy-momentum tensor has a well defined thin
string limit. Furthermore the deficit angle of the conical spacetime that one
obtains in the limit may be given in terms of the distributional
energy-momentum tensor. On the other hand it is only in the special case of
critical coupling that the energy-momentum tensor defined in the Colombeau
algebra is associated to a conventional distribution. The asymptotics of both
the matter and gravitational field are investigated in the thin string limit
and it is shown how this leads to the `conical approximation' which is valid
outside the inner core of the string.
