  We introduce a self-avoiding walk model for which end-effects are completely
eliminated. We enumerate the number of these walks for various lattices in
dimensions two and three, and use these enumerations to study the properties of
this model. We find that endless self-avoiding walks have the same connective
constant as self-avoiding walks, and the same Flory exponent $\nu$. However,
there is no power law correction to the exponential number growth for this new
model, i.e. the critical exponent $\gamma = 1$ exactly. In addition, we have
convincing numerical evidence to support the hypothesis that the amplitude for
the number growth is a universal quantity, and somewhat weaker evidence which
suggests that the number growth has no analytic corrections to scaling. The
technique by which end-effects are eliminated may be generalised to other
models of polymers such as interacting self-avoiding walks.
