diff --git a/src/H5Adense.c b/src/H5Adense.c
index 48004d2aa70..52a6244d7be 100644
--- a/src/H5Adense.c
+++ b/src/H5Adense.c
@@ -839,6 +839,7 @@ H5A__dense_rename(H5F_t *f, const H5O_ainfo_t *ainfo, const char *old_name, cons
     htri_t              attr_sharable;       /* Flag indicating attributes are shareable */
     htri_t              shared_mesg;         /* Should this message be stored in the Shared Message table? */
     bool                attr_exists;         /* Attribute exists in v2 B-tree */
+    H5O_ainfo_t         tainfo    = *ainfo;  /* Copy of ainfo */
     herr_t              ret_value = SUCCEED; /* Return value */
 
     FUNC_ENTER_PACKAGE
@@ -977,8 +978,12 @@ H5A__dense_rename(H5F_t *f, const H5O_ainfo_t *ainfo, const char *old_name, cons
     else if (shared_mesg < 0)
         HGOTO_ERROR(H5E_ATTR, H5E_WRITEERROR, FAIL, "error determining if message should be shared");
 
-    /* Delete old attribute from dense storage */
-    if (H5A__dense_remove(f, ainfo, old_name) < 0)
+    /* Deactivate the field so that H5A__dense_remove() won't delete the new renamed attribute
+       that was just added to the creation order index v2 B-tree via H5A__dense_insert() */
+    tainfo.corder_bt2_addr = HADDR_UNDEF;
+
+    /* Only delete the old attribute (before rename) from the name index v2 B-tree */
+    if (H5A__dense_remove(f, (const H5O_ainfo_t *)&tainfo, old_name) < 0)
         HGOTO_ERROR(H5E_ATTR, H5E_CANTDELETE, FAIL, "unable to delete attribute in dense storage");
 
 done:
diff --git a/test/tattr.c b/test/tattr.c
index d38fdaabc8d..e15ed5fb736 100644
--- a/test/tattr.c
+++ b/test/tattr.c
@@ -2898,6 +2898,11 @@ test_attr_dense_rename(hid_t fcpl, hid_t fapl)
             VERIFY(is_dense, true, "H5O__is_attr_dense_test");
         }
 
+        /* Verify github issue #1388 that the last renamed attribute
+           with/without tracking corder can be deleted */
+        ret = H5Adelete(dataset, new_attrname);
+        CHECK(ret, FAIL, "H5Adelete");
+
         /* Close Dataset */
         ret = H5Dclose(dataset);
         CHECK(ret, FAIL, "H5Dclose");
@@ -2930,8 +2935,8 @@ test_attr_dense_rename(hid_t fcpl, hid_t fapl)
     dataset = H5Dopen2(fid, DSET1_NAME, H5P_DEFAULT);
     CHECK(dataset, H5I_INVALID_HID, "H5Dopen2");
 
-    /* Verify renamed attributes */
-    for (u = 0; u < (max_compact * 2); u++) {
+    /* Verify renamed attributes (the last attribute was deleted) */
+    for (u = 0; u < (max_compact * 2 - 1); u++) {
         unsigned value; /* Attribute value */
 
         /* Open attribute */