SAN FRANCISCO—A jury found that a man developed cancer from exposure to Roundup weedkiller he used in his yard, in the second case to go to trial over the alleged harms of the popular Bayer AG product.

The six-person jury in U.S. District Court in San Francisco now is set to begin hearing evidence to separately weigh whether Bayer’s Monsanto unit should be held liable, a decision that could bring substantial financial damages against the company.

The...