The experiment for cryogenic large-aperture intensity mapping (EXCLAIM) is a balloon-borne telescope designed to survey star formation in windows from the present to z  =  3.5. During this time, the rate of star formation dropped dramatically, while dark matter continued to cluster. EXCLAIM maps the redshifted emission of singly ionized carbon lines and carbon monoxide using intensity mapping, which permits a blind and complete survey of emitting gas through statistics of cumulative brightness fluctuations. EXCLAIM achieves high sensitivity using a cryogenic telescope coupled to six integrated spectrometers employing kinetic inductance detectors covering 420 to 540 GHz with spectral resolving power R  =  512 and angular resolution ≈4  arc min. The spectral resolving power and cryogenic telescope allow the survey to access dark windows in the spectrum of emission from the upper atmosphere. EXCLAIM will survey 305  deg² in the Sloan Digital Sky Survey Stripe 82 field from a conventional balloon flight in 2023. EXCLAIM will also map several galactic fields to study carbon monoxide and neutral carbon emission as tracers of molecular gas. We summarize the design phase of the mission.