  We report the results of a search for the decay $K^+ \to \pi^+ \nu \bar \nu$
in the kinematic region with $\pi^+$ momentum $140 < P < 195$ MeV/c using the
data collected by the E787 experiment at BNL. No events were observed. When
combined with our previous search in this region, one candidate event with an
expected background of $1.22 \pm 0.24$ events results in a 90% C.L. upper limit
of $2.2 \times 10^{-9}$ on the branching ratio of $K^+ \to \pi^+ \nu \bar \nu$.
We also report improved limits on the rates of $K^+ \to \pi^+ X^0$ and $K^+ \to
\pi^+ X^1 X^2$ where $X^0, X^1, X^2$ are hypothetical, massless, long-lived
neutral particles.
