diff --git a/src/requests/adapters.py b/src/requests/adapters.py
index eb240fa954..fc5606bdcb 100644
--- a/src/requests/adapters.py
+++ b/src/requests/adapters.py
@@ -390,6 +390,9 @@ def request_url(self, request, proxies):
             using_socks_proxy = proxy_scheme.startswith("socks")
 
         url = request.path_url
+        if url.startswith("//"):  # Don't confuse urllib3
+            url = f"/{url.lstrip('/')}"
+
         if is_proxied_http_request and not using_socks_proxy:
             url = urldefragauth(request.url)
 
diff --git a/tests/test_adapters.py b/tests/test_adapters.py
new file mode 100644
index 0000000000..6c55d5a130
--- /dev/null
+++ b/tests/test_adapters.py
@@ -0,0 +1,8 @@
+import requests.adapters
+
+
+def test_request_url_trims_leading_path_separators():
+    """See also https://github.com/psf/requests/issues/6643."""
+    a = requests.adapters.HTTPAdapter()
+    p = requests.Request(method="GET", url="http://127.0.0.1:10000//v:h").prepare()
+    assert "/v:h" == a.request_url(p, {})