  We demonstrate that string consistency in four spacetime dimensions leads to
a spectrum of string states which satisfies the supertrace constraints
Str(M^0)=0 and Str(M^2)=\Lambda at tree level, where \Lambda is the one-loop
string cosmological constant. This result holds for a large class of string
theories, including critical heterotic strings. For strings lacking spacetime
supersymmetry, these supertrace constraints will be satisfied as a consequence
of a hidden ``misaligned supersymmetry'' in the string spectrum. These results
thus severely constrain the possible supersymmetry-breaking scenarios in string
theory, and suggest a new intrinsically stringy mechanism whereby such
supertrace constraints may be satisfied without phenomenologically unacceptable
consequences.
