diff --git a/broker/ipc.go b/broker/ipc.go
index 709974b..d0b4d47 100644
--- a/broker/ipc.go
+++ b/broker/ipc.go
@@ -8,13 +8,14 @@ import (
 	"time"
 
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/bridgefingerprint"
+	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/constants"
 
 	"github.com/prometheus/client_golang/prometheus"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/messages"
 )
 
 const (
-	ClientTimeout = 5 // this is calibrated to match the timeout of the CDNs we use for rendezvous
+	ClientTimeout = constants.BrokerClientTimeout
 	ProxyTimeout  = 10
 
 	NATUnknown      = "unknown"
diff --git a/common/constants/constants.go b/common/constants/constants.go
new file mode 100644
index 0000000..ded1659
--- /dev/null
+++ b/common/constants/constants.go
@@ -0,0 +1,10 @@
+package constants
+
+const (
+	// If the broker does not receive the proxy answer within this many seconds
+	// after the broker received the client offer,
+	// the broker will respond with an error to the client.
+	//
+	// this is calibrated to match the timeout of the CDNs we use for rendezvous
+	BrokerClientTimeout = 5
+)
diff --git a/proxy/lib/snowflake.go b/proxy/lib/snowflake.go
index 7bd1aaf..7e761af 100644
--- a/proxy/lib/snowflake.go
+++ b/proxy/lib/snowflake.go
@@ -45,14 +45,13 @@ import (
 	"github.com/pion/transport/v3/stdnet"
 	"github.com/pion/webrtc/v4"
 
+	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/constants"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/event"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/messages"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/namematcher"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/task"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/util"
 	"gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/common/websocketconn"
-
-	snowflakeClient "gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/v2/client/lib"
 )
 
 const (
@@ -540,11 +539,12 @@ func (sf *SnowflakeProxy) makePeerConnectionFromOffer(
 	}
 
 	// Wait for ICE candidate gathering to complete,
-	// or for whatever we managed to gather before the client times out.
+	// or for whatever we managed to gather before the broker
+	// responds with an error to the client offer.
 	// See https://gitlab.torproject.org/tpo/anti-censorship/pluggable-transports/snowflake/-/issues/40230
 	select {
 	case <-done:
-	case <-time.After(snowflakeClient.DataChannelTimeout / 2):
+	case <-time.After(constants.BrokerClientTimeout * time.Second * 3 / 4):
 		log.Print("ICE gathering is not yet complete, but let's send the answer" +
 			" before the client times out")
 	}