Their story
Conrad was of the house of the Guelphs, from the counts of Altorff. He studied at the schools of Constance, was ordained priest and became provost there.
When Bishop Noting died in 934, Conrad was chosen bishop after long reluctance. He was a close friend of St. Ulric of Ausburg. He built and endowed three churches at Constance, dedicated to St. Maurice, St. John the Evangelist and St. Paul. He made three pilgrimages to the holy places at Jerusalem.
Butler tells that one Easter a spider fell into the chalice; Conrad swallowed it out of reverence for the sacrament, and brought it up some hours later unharmed. He died in 976 after 42 years as bishop and was buried at St. Maurice's, where two blind men are said to have been healed. Butler and Baring-Gould agree on 976.