A full-information continuous-time best choice problem is considered. A stream of objects being iid random variables with a known continuous distribution function is observed. The objects appear according to some renewal process independent of objects. The objective is to maximize the probability of selecting of one of the k best objects when observation is perfect, one choice can be made and neither recall nor uncertainty of selection is allowed. The horizon of observation is a positive random variable independent of objects. The natural case of a Poisson renewal process (with intensity [lambda]) and of exponentially distributed horizon (with parameter [mu]) is examined in detail. An optimal stopping rule stops at the first object which is greater than some constant level c(p) depending only on p=[mu]/([mu]+[lambda]). The probability of choosing the proper object P(win) is constant for all natural cases, i.e. when p is small. Simple formulae and numerical values for c(p) and P(win) are obtained. It is interesting that if p tends to 0, P(win) goes to 1 and c(p) goes to 0 at a much slower rate than exponentially fast.
