With some delay, here is the June 2014 CU Release for SharePoint 2013 Server, please find all info below.
All packages except the CU package for SharePoint foundation are now available. The SharePoint foundation package will be available soon, if not already published.

NOTE!
All Updates for SharePoint 2013 requires either the March 2013 PU or SharePoint Server 2013 SP1 as mandatory baseline for all following CU’s !
You must install these updates on RTM Farms before you can install subsequent cumulative updates in the future!

For servers that include search components, you have to follow specific steps to ensure that you install the PU correctly.
Please find here important advise on how to do that:
How to install update packages on a SharePoint farm where search component and high availability search topologies are enabled

Please note the following change for new releases. Previous releases of the SharePoint Server 2013 cumulative update included both the executable and the .CAB file in the same self-extracting executable download. Based on the file size, the SharePoint Server 2013 package has been now divided into several separate downloads. One contains the executable file (identified as i´.e. ubersrv2013kbxxxxxfullfilex64glb), while the other contains the .CAB file (identified as i.e. ubersrv1,…2  etc.). All of them are necessary and must be extracted to the same folder to successfully install the update. All of them are available by clicking the Download link in the KB article for the release.

Example:

Further Information:
As usual, you need to run the SharePoint 2013 Products Configuration Wizard on each machine in the farm after installing the packages.

• The SharePoint Server pack contains the Foundation packs, so you basically only need to install one Package.
• The Project Server Packs are containing both, so if you install this one, no need for the other two to install additional

SharePoint 2013 Updates

Build Number: 15.0.4623.1001
Note:  Farm build number  seen in CA is 15.0.4617.1000 but the kb shows build 15.0.4623.1001.

KB 2881063 – SharePoint Foundation 2013 CU

KB 2881061 – SharePoint Server 2013 CU (aka  Uber package)

KB 2881051 – Office Web Apps Server 2013 (Note: Office Web Apps Server 2013 fixes will be released in different cadence to SharePoint in future)

KB 2881062 – SharePoint Server 2013 with Project Server June 2014 CU

KNOWN ISSUE:

n/a

Related Links:

• How to: install update packages on a SharePoint 2013 farm where search component and high availability search topologies are enabled
• Update center for Office, Office servers, and related products
• Software updates overview for SharePoint 2013
• Project 2013 and Project Server 2013: Language Packs
• CHANGE: SharePoint 2013 Rollup Update for the December 2013 Cumulative Update Packaging

All previous Updates:

• Build Matrix for SharePoint Server 2013
• Build Matrix for SharePoint Server 2010
• Build Matrix for SharePoint Server 2007 (aka MOSS)

Common Questions:

• What is the difference between a PU, a CU and a COD?
• "The expected version of the product was not found on the system"
• SQL Server 2014 and SharePoint Server 2013