We propose a LM-Type statistic to test the null of cointegration allowing for the possibility of a structural break, both in the deterministic and the cointegration vector. The test can be used as a complement of the usual non-cointegration tests in order to get stronger evidence of cointegration. We consider both cases, when the break date is known and when it is not, analysing up to three different procedures to estimate the date of the break. We show that the usual ways to estimate the break date do not produce good results and propose a new procedure that works. The behaviour of the tests is studied through Monte Carlo experiments.