I read about it in a book that my sister gave me, but here are a few tidbits from Wikipedia:
On that very first escort mission, on 24 July 1987, the Kuwaiti oil tanker al-Rekkah, re-flagged as the U.S. tanker MV Bridgeton and accompanied by US navy warships, struck an Iranian underwater mine planted some 20 miles (32 km) west of Farsi Island the night earlier by a Pasdaran special unit, damaging the ship, but causing no injuries. The Bridgeton proceeded under her own power to Kuwait, with the thin-skinned U.S. Navy escorts following behind to avoid mines
On July 24, Bridgeton collided with a mine at a position of 27°58' north and 49°50' east, 13 miles west of Farsi Island. The explosion caused a 43-square-meter dent in the body of the oil tanker. Bridgeton slowed, but did not stop. Meanwhile, the U.S. Navy warships took station in the tanker's wake.
A USN escort also hit a mine and didn't fare as well:
USS Samuel B. Roberts had arrived in the Persian Gulf and was heading for a refueling rendezvous with San Jose on 14 April when the ship struck an M-08 naval mine in the central Persian Gulf, an area she had safely transited a few days earlier. The mine blew a 15 feet (4.6 m) hole in the hull, flooded the engine room, and knocked the two gas turbines from their mounts. The blast also broke the keel of the ship; such structural damage is almost always fatal to most vessels. The crew fought fire and flooding for five hours and saved the ship.
Add some redundant power, comms, propulsion, and CWS defense and it would take an armada to destroy the thing.
It wouldn't move fast or maneuver, but why bother? It's not as though the radar signature would get much smaller.
By definition, a frigate like the USS Samuel B. Roberts lacks the bulk to suffer a mine strike and remain serviceable, while a tanker many times larger may survive a mine strike that misses the vitals of the ship. Different results might obtain in the open sea against anti-ship missiles. There the large tanker would be easy to target and the frigate much harder, with terminal defenses like the Phalanx auto cannon offering a fair chance of destroying an incoming missile.