First, you need a GNU Make >= 3.81
You can look at your version with "make --version"
GNU Make 3.80 _DOES NOT WORK_

Update: fink has make 3.81 now.

To update GNU Make, you can look at
https://sourceforge.net/tracker/index.php?func=detail&aid=1603950&group_id=17203&atid=414256
Download make.patch and make.info
Then, type:
sudo cp make.patch make.info /sw/fink/dists/local/main/finkinfo/
fink index
fink build make
fink install make

Once you have a correct GNU Make, to build and install LaTeX-Make,
you have to :
sudo cp latex-make.info /sw/fink/dists/local/main/finkinfo/latex-make.info
fink index
fink build latex-make
fink install latex-make

Note: as latex-make.info needs to have the md5sum of the source package, it
is not possible to include in a release the latex-make.info corresponding to
this release.
You will always need to get the last latex-make.info from svn...

