Hey,
I was able to reproduce your results in Excel and Origin.
If you are concerned that the Origin results may not be quite right, try using another computational product to reproduce the fitting- for example Wolfram|Alpha (because it's easy to do online):
1) In your browser, navigate to http://http://www.wolframalpha.com/
2) Enter the below text (your data in the W|A required format) into the "Enter what you want to calculate or know about:" text box and click the "=" button.
exponential fit {0.24,1E-03},{0.34,0.005},{0.36,0.01},{0.39,0.02},{0.43,0.05},{0.46,0.1},{0.49,0.2},{0.53,0.5},{0.57,1},{0.6,2},{0.65,5.000},{0.68,10},{0.69,14}
You will see that Origin and W|A agree almost perfectly (as they should).
As per the following reference, Excel does a log transformation of the original Y data and then performs a linear fit of that data to get the fitting parameters for the original Y data. This produces somewhat incorrect results. Read the reference for a more thorough explanation.
http://www.mikemiddleton.com/Excel-Exponential-Curve-Fit-2010.pdf
