Let mu(t)(dx) denote a three-dimensional super-Brownian motion with de
terministic initial state mu(0)(dx) = dx, the Lebesgue measure. Let V:
R(3) --> R be Holder-continuous with compact support, not identically
zero and such that integral(R3)V(x) dx = 0. We show that log P {integ
ral(0)(t) integral(R3)V(x)mu(s)(dx)ds > bt(3/4)} is of order t(1/2) as
t --> infinity, for b > 0. This should be compared with the known res
ult for the case integral(R3)V(x)dx > 0. In that case the normalizatio
n bt(3/4), b > 0, must be replaced by bt, b > integral(R3)V(x)dx, in o
rder that the same statement hold true. While this result only capture
s the logarithmic order, the method of proof enables us to obtain comp
lete results for the corresponding moderate deviations and central lim
it theorems.