Bluebird: Waiting for one promise to settle

At the risk of sounding stupid: What's the most efficient way to wait for one promise to settle? Say I have promise A and I want to create promise B. The fulfillment of B does not depend on the final value of A. It should proceed even if A is rejected. The process shouldn't start, however, until A is settled one way or the other.

What I have currently looks like the following:

var prevResult;

function doStuff() {
    if (prevResult) {
        prevResult = promise.settle([ prevResult ]).then(function() {
            return doStuff();
        })
    } else {
        prevResult = updateDB().finally(function() {
            prevResult = null;
        });
    }
    return prevResult;
}

The code is a bit non-obvious. I'm also a bit worried about chaining a bunch of promises from settle() together, seeing how it's function for performing less trivial kind of coordination. Seems like there ought to be a simpler way of doing this.