This paper analyzes a class of alternating-offer bargaining games with one-sided incomplete information for the case of "no gap." If sequential equilibria are required to satisfy the additional restrictions of stationarity, monotonicity, pure strategies, and no free screening, we establish the Silence Theorem: When the time interval between successive periods is made sufficiently short, the informed party never makes any serious offers in the play of alternating-offer bargaining games. A class of parametric examples suggests that the time interval required to assure silence is not especially brief. As a byproduct of the analysis, we also prove (under the same set of assumptions) a uniform version of the Coase Conjecture: When the time interval between successive periods is made sufficiently short, the initial serious offer by either party in an alternating-offer bargaining game must be less than $\varepsilon$ times the highest possible buyer valuation, for an entire family of distribution functions.