The Catholic World was the publication started by Fr. Hecker in 1865. It was in publication till 1996, when, I believe, financial pressures lead to publication ceasing. Now, as the Society gears up for its 150th anniversary in 2008, the Catholic World has been re-launched online. It's being billed as "an internet journal at the intersection of faith and North American Culture."
The first issue is online. Check it out! Especially the historical piece from 1945 reacting to the use of the bomb. And a piece lambasting George Weigel for his uncharitable column on Cardinal McCarrick in the Denver Catholic Register. And a review (well, more a description than a review) of Nobel Laureate Amartya Sen's new book, "Identity and Violence: The Illusion of Destiny." [I still haven't read the Argumentative Indian which got mixed reviews in the subcontinent and in the West. One of these days ... Here's a review by Tunku Verdarajan at the WSJ.]
Here's the "about us" piece on the Catholic World by Fr. Paul Robichaud, editor.