How to assign new data to an existing clustering

This is the situation. First, I performed factor analysis on 20 continuous variables, and reduced those to 10 factors using SPSS. Then, I did k-means cluster analysis and through trial & error came up with a 6 cluster solution.

What I now need to accomplish is being able to predict which of the existing clusters does my new data belong.

I have read in other forums that I've encountered about using Linear Discriminant Analysis or looking at the Euclidean centroids of the new data points. I don't understand how they get from point A to point B.

Correct me if I'm wrong, but the only thing that makes sense to me is to
  • create a filter or a new data file in SPSS,
  • run Factor Analysis 10 times with my current factors.
  • Then run k-means cluster analysis with those 10 factor variables and 6 clusters. And hope that I don't have any outliers.

Assuming all goes well, append to my original dataset.
This accomplishes, in my mind, saving the centroids, factor means and cluster membership. Because this data set will need to be updated once a month.

Can anyone suggest anything easier?