Let \(A\) be a unital algebra over a commutative unital ring with \(e\in A\) a nontrivial idempotent and set \(f=1-e\). Assume that each of \(eAf\) and \(fAe\) is a faithful module for both \(eAe\) and \(fAf\), on the appropriate sides. A Jordan derivation \(D\) of \(A\) is called `singular' if \(D(eAe)=0=D(fAf)\), \(D(eAf)\subseteq fAe\), and \(D(fAe)\subseteq eAf\). A nonzero singular Jordan derivation of \(A\) cannot be a derivation of \(A\). The main result of the authors shows that any Jordan derivation \(D\) of \(A\) is a sum of a derivation of \(A\) and a singular Jordan derivation of \(A\), each uniquely determined by \(D\). Two known results that follow from this are that a Jordan derivation of a triangular algebra, or of a prime unital algebra with idempotent, must be a derivation.