 On 28 March 1977 Konstant was appointed as an auxiliary bishop of the Roman Catholic Archdiocese of Westminster and Titular Bishop of Betagbarar under the papacy of Pope Paul VI.On 25 April 1977, he was consecrated bishop, the principal consecrator being Cardinal Basil Hume, along with auxiliary bishops Gerald Mahon and Victor Guazzelli as co-consecrators.On 12 July 1985, Konstant was named as Bishop of the Roman Catholic Diocese of Leeds and was installed as the eighth Bishop of Leeds on 25 September 1985 under the papacy of Pope John Paul II.