The LM317 has a minimum worst-case output current of 10mA so it will not reliably deliver only 1mA as a constant-current source.
Don't know of a better IC.
One common approach is to use a voltage reference, an op amp and a MOSFET connected as a constant-current source (see below). The op amp can be any rail-to-rail type and the transistor can be just about any logic-level type P-MOSFET. The reference is a two-terminal 1.2V shunt-type regulator.
The op amp feedback adjusts the transistor gate voltage to drop 1.2V across the source resistor R1, thus generating a constant current at the drain output through the PT resistance, Rpt.
Edit: The LT1009 actually has a drop of about 1.235V nominal so R1 should be 1.235k ohm (1.24K is the closest 1% value) to give 1.0mA current.
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