The point of instability of a Bose-Einstein condensate (BEC) due to attract
ive interactions was studied. Stable Rb-85 BECs were created and then cause
d to collapse by slowly changing the atom-atom interaction from repulsive t
o attractive using a Feshbach resonance. At a critical value, an abrupt tra
nsition was observed in which atoms were ejected from the condensate. By me
asuring the onset of this transition as a function of number and attractive
interaction strength, we determined the stability condition to be N\a\/a(h
o) = 0.459 +/- 0.012 +/- 0.054, slightly lower than the predicted Value of
0.574.