Web Service Returns 503 Error

[HBrothers]

I have a web service running on W2003, IIS 6.0 with a client program running
on XP sp 2.
The web service is called often and repeatedly by the client program, and it
spends some time inserting rows into a SQL database on another server.

Occasionally, the web service returns this error message to the client:
The request failed with HTTP status 503: Service Unavailable

I have tried running with sessions enabled and disabled in the Web service,
with a single proxy object or with a new proxy object created on each call
from the client program, etc. I cannot predict ahead of time how many calls
the client will make to the web service -- it depends on the number of lines
in a text file that will be different with each invocation of the program.

The IIS log contains a reference to the .asmx page containing the web
service followed by
503 537842195 ConnLimit

I gather from all of this that the web service or the client program (or
both) are mis-managing resources somehow, but I don't see how to fix the
situation. Any help would be very much appreciated.