A threshold version of a CMOS 4050?
You should use a comparator for this (or a single rail op-amp).  

Most digital logic gates require more than 30% of VCC to trigger reliably and not burn more static power.  You could check some datasheets, if you wanted.  The key term you are looking for is \$ V_{IH} \$ (for input-high).  This represents the lowest possible input which will register as a clean '1', or high logic.  For most 4000 series IC's, this value is ~4V.  The \$ V_{IL} \$ is likewise the highest input voltage which will register as a clean '0'.  For CMOS 4000 series, this is ~1V.

The difference between these values is ~3V.  Normally, higher is better, because this means you would require more noise to cause an output error.  The situation you described, with \$V_{IL} = 1V\$ and \$V_{IH} = 1.5V\$ only leaves 0.5V for noise.  So for general purpose gates, most will not have these characteristics.  

4000 series datasheet

But, you can use a comparator (or just plain old single rail op-amp) with the V- = 1.5V and the V+ as your input.  You can give the comparator IC supply and ground of +5V and 0V, so it will produce your 0-5V output signal. (Like LM293)

Edit: 
As pointed out by Peter Bennett, the LM293 has an open collector output.  So connect the output to the supply voltage with ~3.3kOhm or so resistor.  Most op-amps do not require this pullup resistor; just check the datasheets.