We address the question of the stability of newly formed stellar tripl
e systems that are undergoing accretion, Accretion of infalling gas ca
n help stabilize unstable triple systems and can destabilize stable on
es. The dynamics of the accretion depend critically on the specific an
gular momentum of the infalling gas relative to that of the triple sys
tem and on the component masses. Low relative specific angular momentu
m accretion helps to stabilize systems where a lower mass single star
is orbiting a more massive binary, by preferentially shrinking the sep
aration of the binary over that of the binary-single. High relative sp
ecific angular momentum accretion also helps to stabilize these system
s, but by increasing the binary-single separation. Moderate values of
the relative specific angular momentum of the infalling matter can des
tabilize previously stable systems, Systems with comparable-mass singl
e components are stabilized by accreting matter with significant angul
ar momentum, while they can be destabilized by accreting zero angular
momentum material. Triples composed of a binary orbiting a more massiv
e star are easily destabilized for most values of the specific angular
momentum of the infalling gas. Such systems are unlikely to survive s
ignificant accretion, We comment on how these scenarios can arise depe
nding on the triple formation mechanism, and discuss the implications
that accretion has for stellar multiplicity.