VATICAN CITY — With a humble ‘‘Grazie’’ as bishops doffed their mitres and applause echoed through St. ­Peter’s Basilica, a frail Pope Benedict XVI began his long farewell by presiding over Ash Wednesday services in a tearful, final public Mass.

‘‘We wouldn’t be sincere, Your Holiness, if we didn’t tell you that there’s a veil of sadness on our hearts this evening,’’ said Cardinal Tarcisio Bertone, Benedict’s longtime deputy, his voice breaking.