The interval estimation design is studied for a second-order delay differential equation with position delayed measurements, uncertain input and initial conditions. The proposed method contains two consecutive interval observers. The first one estimates the interval of admissible values for the position without delay for each instant of time using new delay-dependent conditions on positivity. Then derived interval estimates of the position are used to design the second observer estimating an interval of admissible values for the velocity of the considered dynamical system. The results are illustrated by numerical experiments for an example.