In the present paper, we show that the Morimoto Conjecture on the super additivity of the tunnel numbers of knots in
S
3
S^3
is true for knots
K
1
,
K
2
K_1,K_2
in
S
3
S^3
in which each
K
i
K_i
is either a tunnel number one or
m
m
-small,
i
=
1
,
2
i=1,2
. This extends two known results by Morimoto.