Pope Francis has confirmed the appointment, which will see Stock become the city's tenth bishop.
The position became vacant after archbishop Arthur Roche took up the position of secretary of the Congregation for Divine Worship and the Discipline of the Sacraments in Rome.
Bishop-elect Stock, 53, was ordained as a priest in 1988 and served as parish priest across the Archdiocese of Birmingham, most recently at the parish of The Sacred Heart and St Teresa in Coleshill.
He is also director of the Diocesan Schools' Commission and has been general secretary of the Catholic Bishops' Conference.
He said: 'I am overwhelmed at the trust which the Holy Father has placed in me.
'I have written to the Holy Father and assured him, as the successor of St Peter and supreme pontiff of the Catholic Church, of my loyal and filial love, and of my intention to give myself entirely to the service of the Diocese of Leeds.
'I am only too conscious of my weaknesses and sinfulness. It is therefore with humility and the desire to obey the call to serve, in whatever way the Church asks of me, that I have accepted this appointment.
'Before all else, therefore, I ask for your prayers for me. Throughout my ministry as a priest, I have been sustained in grace by the prayers of the parishioners and religious that I have been privileged to serve and by the prayers of my brother clergy in the Archdiocese of Birmingham. This is the first thing for which I hope; to be assured of your prayers.
'And from now on, I assure you, for my part you will be in the first thoughts of my prayers each day.'
Cardinal Vincent Nichols said: "For the last five years I have been constantly grateful to the Archdiocese of Birmingham for releasing Mgr Marcus Stock to be the general secretary of the Bishops' Conference.
'As he is now appointed to be the next Bishop of Leeds, I express my warmest thanks to Mgr Stock for the great work he has done here in London on behalf of us bishops and the Church.
'The Diocese of Leeds is receiving a fine man as its new bishop. He will bring his many abilities to his new calling. They are well known and widely appreciated. Most importantly of all, he brings his deep love of the Lord and his Church; the true source of his dedication, sensitivity, generosity and stability.'