  Poincar\'e-invariant quantum field theories can be formulated on
non-commutative planes if the coproduct on the Poincar\'e group is suitably
deformed \cite{Dimitrijevic:2004rf, Chaichian:2004za}.(See also especially
Oeckl \cite{Oeckl:1999jun},\cite{Oeckl:2000mar} and Grosse et
al.\cite{Grosse:2001mar}) As shown in \cite{Balachandran:2005eb}, this
important result of these authors implies modification of free field
commutation and anti-commutation relations and striking phenomenological
consequences such as violations of Pauli principle
\cite{Balachandran:2005eb,Bal3}. In this paper we prove that with these
modifications, UV-IR mixing disappears to all orders in perturbation theory
from the $S$-Matrix. This result is in agreement with the previous results of
Oeckl \cite{Oeckl:2000mar}.
