You might try one of these two approaches. Both of them are based on the logit transform of the response Y. The logit expects an argument [0, 1] for the constraint you want but produces a response [-infinity, infinity] that regression requires.
I mocked up some data to illustrate this approach.
Apply the Logit transform to the response Y in the Fit Model dialog. Select Analyze > Fit Model and enter the data columns in the respective analysis roles. Select the column in the Y role, then click the red triangle next to Transform and select Logit.
Your result will be familiar, you can use the Profiler for example, but the constraint should apply.
The other approach is to use a Generalized Linear Model. Start as before but do not apply the transformation. Instead, click Personality and select Generalized Linear Model. Click Distribution and select Normal. Click Link and select Logit.