  Often in sequential trials additional data become available after a stopping
boundary has been reached. A method of incorporating such information from
overrunning is developed, based on the ``adding weighted Zs'' method of
combining $p$-values. This yields a combined $p$-value for the primary test and
a median-unbiased estimate and confidence bounds for the parameter under test.
When the amount of overrunning information is proportional to the amount
available upon terminating the sequential test, exact inference methods are
provided; otherwise, approximate methods are given and evaluated. The context
is that of observing a Brownian motion with drift, with either linear stopping
boundaries in continuous time or discrete-time group-sequential boundaries. The
method is compared with other available methods and is exemplified with data
from two sequential clinical trials.
