Jesse's Ex Threatened to Kill Him -- Allegedly

Jesse James' ex-wife, Janine Lindemulder, threatened to kill him ... this according to what Jesse's lawyer just told the judge in their custody war.

Jesse has been testifying on the stand this morning, but hasn't mentioned the allegation ... so far.

As we previously reported, Jesse bought a house in Austin and wants to take Sunny there so she can be closer to Sandra Bullock ... who developed a close relationship with Sunny before her divorce from Jesse.

But Janine filed documents trying to stop the move -- saying the Sandra excuse is BS because if she really cared about Jesse's kids ... she would have stayed in California.

Story developing...