  We use Arakelov theory to define a height on divisors of degree zero on a
hyperelliptic curve over a global field, and show that this height has
computably bounded difference from the N\'eron-Tate height of the corresponding
point on the Jacobian. We give an algorithm to compute the set of points of
bounded height with respect to this new height. This provides an `in principle'
solution to the problem of determining the sets of points of bounded
N\'eron-Tate heights on the Jacobian. We give a worked example of how to
compute the bound over a global function field for several curves, of genera up
to 11.
