#given an original pandas.DataFrame named 'df', and the data series that we want to associate with the new column named 's', 3 possible solutions:

#1-Use "assign()" to create new column and then assign values to it
df_w_new_col = df.assign (name='New_Col')

#2-create an aditional Dataframe with just the new column to add and concatenate with the old dataframe. Like a great majority of pandas methods, this actually creates a new df, it does not just concatenate to the old one so it is safe to change the old and new df's independetly
df = pd.concat( [df, pd.DataFrame({'New_Col:s})], axis=1 )

#3-explicitely create a copy and append a column
df2 = df.copy()
df2['New_Col'] = s

In [483]: A.assign(name=s)
Out[483]:
          0         1  name
0  0.876880  0.915174     0
1  0.644664  0.305387     1
2  0.930093  0.562453     2
3  0.807626  0.498871     3