Johnson & Johnson was ordered by a jury to pay $300 million in punitive damages to a woman who blamed her rare asbestos-related cancer on decades of daily use of the company’s talc-based products.

The ruling brings to $325 million the amount the state-court jury in Manhattan said J&J should pay Donna Olson and her husband over her cancer that she blamed on J&J’s baby powder and its former Shower-to-Shower product.

The total verdict is one of the highest in...