  The critical current density flowing across low angle grain boundaries in
YBa$_2$Cu$_3$O$_{7-\delta}$ thin films has been studied magnetometrically.
Films (200 nm thickness) were deposited on SrTiO$_3$ bicrystal substrates
containing a single [001] tilt boundary, with angles of 2, 3, 5, and 7 degrees,
and the films were patterned into rings. Their magnetic moments were measured
in applied magnetic fields up to 30 kOe at temperatures of 5 - 95 K; current
densities of rings with or without grain boundaries were obtained from a
modified critical state model. For rings containing 5 and 7 degree boundaries,
the magnetic response depends strongly on the field history, which arises in
large part from self-field effects acting on the grain boundary.
