Afghanistan, Aug. 19 -- A suicide bombing at a wedding party in Kabul claimed by a local Islamic State affiliate has renewed fears about the growing threat posed by its thousands of fighters, as well as their ability to plot global attacks from a stronghold in the forbidding mountains of northeastern Afghanistan.

The attack came as the Taliban appear to be nearing a deal with the U.S. to end nearly 18 years of fighting. Now Washington hopes the Taliban can help rein in IS fighters, even as some worry that Taliban fighters, disenchanted by a peace deal, could join IS.

The U.S. envoy in talks with the Taliban, Zalmay Khalilzad, says the peace process must be accelerated to put Afghanistan in a "much stronger position to defeat" the Islami...