  Let C in C_1xC_2 be a curve of type (d_1,d_2) in the product of the two
curves C_1 and C_2. Let d be a positive integer. We prove that if a certain
inequality involving d_1, d_2, d, and the genera of the curves C_1, C_2, and C
is satisfied, then the set of points P in C(\kbar) with [k(P):k]<=d is finite
for any number field k. We prove a similar result for integral points of
bounded degree on C. These results are obtained as consequences of an
inequality of Vojta which generalizes the Roth-Wirsing theorem to curves.
