Step 1: List the statements.
Some Professors are Doctors, and All Doctors are Patients.
Step 2: Draw the overlap.
There is a group that is both Professor and Doctor; that group is non-empty because of the word some.
Step 3: Push the overlap into Patients.
Since all Doctors are Patients, every Professor who is a Doctor is also a Patient.
Step 4: Evaluate Conclusion I.
That gives at least one Professor who is a Patient, so Some Professors are Patients definitely follows.
Step 5: Evaluate Conclusion II.
Conclusion II says No Doctor is a Professor, but the first statement already guarantees some are both, so II directly contradicts the data and cannot follow.
Step 6: Choose the option.
Only Conclusion I follows.
\[ \boxed{\text{Only Conclusion I follows.}} \]