This paper states new sufficient conditions for the exponential stability of linear time-varying (LTV) systems of the form $\dot x(\cdot) = A(t) x(t)$. The approach proposed derives and uses the notion of perturbed frozen time (PFT) form that can be associated to any LTV system. Exploiting the Bellman-Gronwall lemma, relaxed stability conditions are then stated in terms of "average" parameter variations. Salient features of the approach are: pointwise stability of $A(\cdot)$ is not required, $\|\dot A(\cdot)\|$ may not be bounded, the stability conditions also apply to uncertain systems. The approach is illustrated by numerical examples.