First I modified php_functions.c and changed "MODULE_MAGIC_AT_LEAST" to "AP_MODULE_MAGIC_AT_LEAST"
Then I proceeded with building:

./configure --with-apxs2=/usr/local/apache2/bin/apxs
make

This eventually gave the following errors:

sapi_apache2.c: In function `php_register_hook':
sapi_apache2.c:534: warning: passing arg 3 of `ap_register_output_filter' makes pointer from integer without a cast
sapi_apache2.c:534: too few arguments to function `ap_register_output_filter'
sapi_apache2.c:535: warning: passing arg 3 of `ap_register_input_filter' makes pointer from integer without a cast
sapi_apache2.c:535: too few arguments to function `ap_register_input_filter'

I tracked this down to a change made recently in Apache's include file util_filter.h.  The older functions are defined as:

AP_DECLARE(ap_filter_rec_t *) ap_register_input_filter(const char *name, ap_in_filter_func filter_func, ap_filter_type ftype);

AP_DECLARE(ap_filter_rec_t *) ap_register_output_filter(const char *name, ap_out_filter_func filter_func, ap_filter_type ftype);

The newer functions have an additional argument:

AP_DECLARE(ap_filter_rec_t *) ap_register_input_filter(const char *name, ap_in_filter_func filter_func, ap_init_filter_func filter_init, ap_filter_type ftype);

AP_DECLARE(ap_filter_rec_t *) ap_register_output_filter(const char *name, ap_out_filter_func filter_func, ap_init_filter_func filter_init, ap_filter_type ftype);

Not sure what the appropriate fix is.  I just added a NULL to the third argument in the function call to sapi_apache2filter.c and it compiles ok.  Here's a diff.

--- sapi/apache2filter/sapi_apache2.c.orig      Mon May  6 05:42:23 2002
+++ sapi/apache2filter/sapi_apache2.c   Fri Aug 30 17:53:22 2002
@@ -531,8 +531,8 @@
        ap_hook_post_config(php_apache_server_startup, NULL, NULL, APR_HOOK_MIDDLE);
        ap_hook_insert_filter(php_insert_filter, NULL, NULL, APR_HOOK_MIDDLE);
     ap_hook_post_read_request(php_post_read_request, NULL, NULL, APR_HOOK_MIDDLE);
-       ap_register_output_filter("PHP", php_output_filter, AP_FTYPE_RESOURCE);
-       ap_register_input_filter("PHP", php_input_filter, AP_FTYPE_RESOURCE);
+       ap_register_output_filter("PHP", php_output_filter, NULL, AP_FTYPE_RESOURCE);
+       ap_register_input_filter("PHP", php_input_filter, NULL, AP_FTYPE_RESOURCE);
 }

 AP_MODULE_DECLARE_DATA module php4_module = {

Apache starts up but doesn't process any PHP code.  Any hints would be appreciated.