Save Arms control and Verification Efforts Act of 2019 or SAVE Act This bill requires reports related to extension of the Measures for the Further Reduction and Limitation of Strategic Offensive Arms (New START Treaty) and prohibits increases in the number of certain U.S. weapons should the treaty lapse. The following reports to Congress shall be required if the parties to the treaty do not finalize a treaty extension: separate reports from the President, the Department of Defense, and the Department of State justifying why the treaty has not been extended and certifying that it is in national security interests to not extend; and an assessment by the Office of the Director of National Intelligence of why the treaty has not been extended, including a certification that the United States will not lose intelligence into Russia's nuclear program. The bill also requires reports to Congress on the effect of a treaty lapse on U.S. intelligence-collection capabilities, the structure of the Armed Forces, U.S. nuclear weapons programs, and foreign relations. The President shall report to Congress on the progress of treaty-related discussions with Russia and a strategy for future strategic arms control agreements with Russia. The bill prohibits increasing the number of various U.S. weapons (such as deployed intercontinental ballistic missiles) above specified thresholds if the treaty lapses or if the President takes certain actions to withdraw from or suspend the treaty, unless the President certifies that Russia has made certain militarily significant increases to its weapons stockpiles.