We introduce a pivot for exact selective inference with randomization. Not only does our pivot lead to exact inference in Gaussian regression models, but it is also available in closed form. We reduce the problem of exact selective inference to a bivariate truncated Gaussian distribution. By doing so, we give up some power that is achieved with approximate maximum likelihood estimation in Panigrahi and Taylor (2022). Yet our pivot always produces narrower confidence intervals than a closely related data splitting procedure. We investigate the trade-off between power and exact selective inference on simulated datasets and an HIV drug resistance dataset.