  We investigate some simple and surprising properties of a one-dimensional
Brownian trajectory with diffusion coefficient $D$ that starts at the origin
and reaches $X$ either: (i) at time $T$ or (ii) for the first time at time $T$.
We determine the most likely location of the first-passage trajectory from
$(0,0)$ to $(X,T)$ and its distribution at any intermediate time $t<T$. A
first-passage path typically starts out by being repelled from its final
location when $X^2/DT\ll 1$. We also determine the distribution of times when
the trajectory first crosses and last crosses an arbitrary intermediate
position $x<X$. The distribution of first-crossing times may be unimodal or
bimodal, depending on whether $X^2/DT\ll 1$ or $X^2/DT\gg 1$. The form of the
first-crossing probability in the bimodal regime is qualitatively similar to,
but more singular than, the well-known arcsine law.
