Conrad Hilton was just formally sentenced in his grand theft auto case -- the same one in which he went off on a racist and homophobic rant.

Conrad was in L.A. County Court Wednesday where he pled no contest to 1 count of felony grant theft auto and 1 count of misdemeanor contempt of court. He was sentenced to 3 years probation and was ordered to continue with mental health and substance counseling.

Hilton violated a restraining order last year against E.G. Daily's daughter, after he allegedly stole her dad's car and drove over to her house. That led to a meltdown with cops.

Hilton was also ordered to stay away from Daily and family for the next 3 years.

As we reported ... Conrad was caught on camera in an incredibly racist and homophobic tirade against cops who arrested him at E.G.'s pad last spring. He screamed homophobic epithets, rape ... and the n-word. It was clear he wasn't well during this time.